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  • Maximum degree and diversity in intersecting hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-15
    Peter Frankl

    Let S be an n-element set and F⊂(Sk) an intersecting family. Improving earlier results it is proved that for n>72k there is an element of S that is contained in all but (n−3k−2) members of F. One of the main ingredients of the proof is the following statement. If G⊂(Sk) is intersecting, |G|≥(n−2k−2) and n≥72k then there is an element of S that is contained in more than half of the members of G.

    更新日期:2020-01-16
  • The EKR property for flag pure simplicial complexes without boundary
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-14
    Jorge Alberto Olarte; Francisco Santos; Jonathan Spreer; Christian Stump

    We prove that the family of facets of a pure simplicial complex C of dimension up to three satisfies the Erdős-Ko-Rado property whenever C is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.

    更新日期:2020-01-15
  • Chordality, d-collapsibility, and componentwise linear ideals
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-14
    Mina Bigdeli; Sara Faridi

    Using the concept of d-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of “chordal clutters” which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with a linear resolution in a polynomial ring. We show d-collapsible and d-representable complexes produce componentwise linear ideals for appropriate d. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees. We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and square-free stable ideals have chordal Stanley-Reisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the Stanley-Reisner ideal of a chordal complex.

    更新日期:2020-01-15
  • On the ℓ4:ℓ2 ratio of functions with restricted Fourier support
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-14
    Naomi Kirshner; Alex Samorodnitsky

    Given a subset A⊆{0,1}n, let μ(A) be the maximal ratio between ℓ4 and ℓ2 norms of a function whose Fourier support is a subset of A.1 We make some simple observations about the connections between μ(A) and the additive properties of A on one hand, and between μ(A) and the uncertainty principle for A on the other hand. One application obtained by combining these observations with results in additive number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining μ(A) rather precisely, when A is a Hamming sphere S(n,k) for all 0≤k≤n.

    更新日期:2020-01-15
  • On the interplay between additive and multiplicative largeness and its combinatorial applications
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-13
    Vitaly Bergelson; Daniel Glasscock

    Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of N and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any n,k∈N, any finite set with fewer than n elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero kth power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of N and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of N of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemerédi and van der Waerden on arithmetic progressions.

    更新日期:2020-01-14
  • Hamiltonicity in randomly perturbed hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-10
    Jie Han; Yi Zhao

    For integers k≥3 and 1≤ℓ≤k−1, we prove that for any α>0, there exist ϵ>0 and C>0 such that for sufficiently large n∈(k−ℓ)N, the union of a k-uniform hypergraph with minimum vertex degree αnk−1 and a binomial random k-uniform hypergraph G(k)(n,p) with p≥n−(k−ℓ)−ϵ for ℓ≥2 and p≥Cn−(k−1) for ℓ=1 on the same vertex set contains a Hamiltonian ℓ-cycle with high probability. Our result is best possible up to the values of ϵ and C and answers a question of Krivelevich, Kwan and Sudakov.

    更新日期:2020-01-11
  • Toeplitz minors and specializations of skew Schur polynomials
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-06
    David García-García; Miguel Tierz

    We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.

    更新日期:2020-01-07
  • 3D positive lattice walks and spherical triangles
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-06
    B. Bogosel; V. Perrollaz; K. Raschel; A. Trotignon

    In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard decomposition, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision.

    更新日期:2020-01-07
  • New necessary conditions on (negative) Latin square type partial difference sets in abelian groups
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-07
    Zeying Wang

    Partial difference sets (for short, PDSs) with parameters (n2, r(n−ϵ), ϵn+r2−3ϵr, r2−ϵr) are called Latin square type (respectively negative Latin square type) PDSs if ϵ=1 (respectively ϵ=−1). In this paper, we will give restrictions on the parameter r of a (negative) Latin square type partial difference set in an abelian group of non-prime power order a2b2, where gcd⁡(a,b)=1, a>1, and b is an odd positive integer ≥3. Very few general restrictions on r were previously known. Our restrictions are particularly useful when a is much larger than b. As an application, we show that if there exists an abelian negative Latin square type PDS with parameter set (9p4s,r(3p2s+1),−3p2s+r2+3r,r2+r), 1≤r≤3p2s−12, p≡1(mod4) a prime number and s is an odd positive integer, then there are at most three possible values for r. For two of these three r values, J. Polhill gave constructions in 2009 [10].

    更新日期:2020-01-07
  • A generalized Eulerian triangle from staircase tableaux and tree-like tableaux
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-07
    Bao-Xuan Zhu

    Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array [Tn,k]n,k≥0, which satisfies the recurrence relation:Tn,k=λ(a1k+a2)Tn−1,k+[(b1−da1)n−(b1−2da1)k+b2−d(a1−a2)]Tn−1,k−1+d(b1−da1)λ(n−k+1)Tn−1,k−2, where T0,0=1 and Tn,k=0 unless 0≤k≤n. We derive some properties of [Tn,k]n,k≥0, including the explicit formulae of Tn,k and the exponential generating function of the generalized Eulerian polynomial Tn(q), and the ordinary generating function of Tn(q) in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of Tn(q), stability of the iterated Turán-type polynomial Tn+1(q)Tn−1(q)−Tn2(q). Furthermore, we also prove the q-Stieltjes moment property and 3-q-log-convexity of Tn(q) and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for γ-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get γ-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types A and B in a unified manner. We also derive γ-positivity for a symmetric sub-array of [Tn,k]n,k≥0, which in particular gives a unified proof of γ-positivity of Eulerian polynomials of types A and B. Our results not only can immediately apply to Eulerian triangles of two kinds and arrays from staircase tableaux and tree-like tableaux, but also to segmented permutations and flag excedance numbers in colored permutations groups in a unified approach. In particular, we also confirm a conjecture of Nunge about the unimodality from segmented permutations.

    更新日期:2020-01-07
  • On cubic graphical regular representations of finite simple groups
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-06-13
    Binzhou Xia

    A recent conjecture of the author and Teng Fang states that there are only finitely many finite simple groups with no cubic graphical regular representation. In this paper, we make crucial progress towards this conjecture by giving an affirmative answer for groups of Lie type of large rank.

    更新日期:2020-01-04
  • On 1-factors with prescribed lengths in tournaments
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-07-03
    Dong Yeap Kang; Jaehoon Kim

    We prove that every strongly 1050t-connected tournament contains all possible 1-factors with at most t components and this is best possible up to constant. In addition, we can ensure that each cycle in the 1-factor contains a prescribed vertex. This answers a question by Kühn, Osthus, and Townsend. Indeed, we prove more results on partitioning tournaments. We prove that a strongly Ω(k4tq)-connected tournament admits a vertex partition into t strongly k-connected tournaments with prescribed sizes such that each tournament contains q prescribed vertices, provided that the prescribed sizes are Ω(n). This result improves the earlier result of Kühn, Osthus, and Townsend. We also prove that for a strongly Ω(t)-connected n-vertex tournament T and given 2t distinct vertices x1,…,xt,y1,…,yt of T, we can find t vertex disjoint paths P1,…,Pt such that each path Pi connecting xi and yi has the prescribed length, provided that the prescribed lengths are Ω(n). For both results, the condition of connectivity being linear in t is best possible, and the condition of prescribed sizes being Ω(n) is also best possible.

    更新日期:2020-01-04
  • The complexity of perfect matchings and packings in dense hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-07-19
    Jie Han; Andrew Treglown

    Given two k-graphs H and F, a perfect F-packing in H is a collection of vertex-disjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F-packing is simply a perfect matching. For a given fixed F, it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F-packing is NP-complete. Indeed, if k≥3, the corresponding problem for perfect matchings is NP-complete [17], [7] whilst if k=2 the problem is NP-complete in the case when F has a component consisting of at least 3 vertices [14]. In this paper we give a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect F-packings is polynomial time solvable. We then give three applications of this tool: (i) Given 1≤ℓ≤k−1, we give a minimum ℓ-degree condition for which it is polynomial time solvable to determine whether a k-graph satisfying this condition has a perfect matching; (ii) Given any graph F we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect F-packing; (iii) We also prove a similar result for perfect K-packings in k-graphs where K is a k-partite k-graph. For a range of values of ℓ,k (i) resolves a conjecture of Keevash, Knox and Mycroft [20]; (ii) answers a question of Yuster [47] in the negative; whilst (iii) generalises a result of Keevash, Knox and Mycroft [20]. In many cases our results are best possible in the sense that lowering the minimum degree condition means that the corresponding decision problem becomes NP-complete.

    更新日期:2020-01-04
  • On Frank's conjecture on k-connected orientations
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-07-10
    Olivier Durand de Gevigney

    We disprove a conjecture of Frank [3] stating that each weakly 2k-connected graph has a k-vertex-connected orientation. For k≥3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

    更新日期:2020-01-04
  • 7-Connected graphs are 4-ordered
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-07-11
    Rose McCarty; Yan Wang; Xingxing Yu

    A graph G is k-ordered if for any distinct vertices v1,v2,…,vk∈V(G), it has a cycle through v1,v2,…,vk in order. Let f(k) denote the minimum integer so that every f(k)-connected graph is k-ordered. The first non-trivial case of determining f(k) is when k=4, where the previously best known bounds are 7≤f(4)≤40. We prove that in fact f(4)=7.

    更新日期:2020-01-04
  • On a conjecture of Bondy and Vince
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-07-25
    Jun Gao; Jie Ma

    Twenty years ago Bondy and Vince conjectured that for any nonnegative integer k, except finitely many counterexamples, every graph with k vertices of degree less than three contains two cycles whose lengths differ by one or two. The case k≤2 was proved by Bondy and Vince, which resolved an earlier conjecture of Erdős et al. In this paper we confirm this conjecture for all k.

    更新日期:2020-01-04
  • A refinement of choosability of graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-08-06
    Xuding Zhu

    Assume k is a positive integer, λ={k1,k2,…,kq} is a partition of k and G is a graph. A λ-assignment of G is a k-assignment L of G such that the colour set ⋃v∈V(G)L(v) can be partitioned into q subsets C1∪C2…∪Cq and for each vertex v of G, |L(v)∩Ci|=ki. We say G is λ-choosable if for each λ-assignment L of G, G is L-colourable. It follows from the definition that if λ={k}, then λ-choosable is the same as k-choosable, if λ={1,1,…,1}, then λ-choosable is equivalent to k-colourable. For the other partitions of k sandwiched between {k} and {1,1,…,1} in terms of refinements, λ-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions λ,λ′ of k, every λ-choosable graph is λ′-choosable if and only if λ′ is a refinement of λ. Then we study λ-choosability of special families of graphs. The Four Colour Theorem says that every planar graph is {1,1,1,1}-choosable. A very recent result of Kemnitz and Voigt implies that for any partition λ of 4 other than {1,1,1,1}, there is a planar graph which is not λ-choosable. We observe that, in contrast to the fact that there are non-4-choosable 3-chromatic planar graphs, every 3-chromatic planar graph is {1,3}-choosable, and that if G is a planar graph whose dual G⁎ has a connected spanning Eulerian subgraph, then G is {2,2}-choosable. We prove that if n is a positive even integer, λ is a partition of n−1 in which each part is at most 3, then Kn is edge λ-choosable. Finally we study relations between λ-choosability of graphs and colouring of signed graphs and generalized signed graphs. A conjecture of Máčajová, Raspaud and Škoviera that every planar graph is signed 4-colcourable is recently disproved by Kardoš and Narboni. We prove that every signed 4-colourable graph is weakly 4-choosable, and every signed Z4-colourable graph is {1,1,2}-choosable. The later result combined with the above result of Kemnitz and Voigt disproves a conjecture of Kang and Steffen that every planar graph is signed Z4-colourable. We shall show that a graph constructed by Wegner in 1973 is also a counterexample to Kang and Steffen's conjecture, and present a new construction of a non-{1,3}-choosable planar graphs.

    更新日期:2020-01-04
  • Linear min-max relation between the treewidth of an H-minor-free graph and its largest grid minor
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-08-13
    Ken-ichi Kawarabayashi; Yusuke Kobayashi

    A key theorem in algorithmic graph-minor theory is a min-max relation between the treewidth of a graph (i.e., the minimum width of a tree-decomposition) and the maximum size of a grid minor. This min-max relation is a keystone of the graph minor theory of Robertson and Seymour, which ultimately proves Wagner's Conjecture about properties of minor-closed graphs. Demaine and Hajiaghayi proved a remarkable linear min-max relation for graphs excluding any fixed minor H: every H-minor-free graph of treewidth at least cHr has an r×r-grid minor for some constant cH. However, as they pointed out, a major issue with this theorem is that their proof heavily depends on the graph minor theory, most of which lacks explicit bounds and is believed to have very large bounds. Motivated by this problem, we give another (relatively short and simple) proof of this result without using the machinery of the graph minor theory. Hence we give an explicit bound for cH, which is an exponential function of a polynomial in |H|. Furthermore, our result gives a constant w=2O(r2log⁡r) such that every graph of treewidth at least w has an r×r-grid minor.

    更新日期:2020-01-04
  • A local epsilon version of Reed's Conjecture
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-08-28
    Tom Kelly; Luke Postle

    In 1998, Reed conjectured that every graph G satisfies χ(G)≤⌈12(Δ(G)+1+ω(G))⌉, where χ(G) is the chromatic number of G, Δ(G) is the maximum degree of G, and ω(G) is the clique number of G. As evidence for his conjecture, he proved an “epsilon version” of it, i.e. that there exists some ε>0 such that χ(G)≤(1−ε)(Δ(G)+1)+εω(G). It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a “local version” of the list-coloring version of Reed's conjecture. Namely, we conjecture that if G is a graph with list-assignment L such that for each vertex v of G, |L(v)|≥⌈12(d(v)+1+ω(v))⌉, where d(v) is the degree of v and ω(v) is the size of the largest clique containing v, then G is L-colorable. Our main result is that an “epsilon version” of this conjecture is true, under some mild assumptions. Using this result, we also prove a significantly improved lower bound on the density of k-critical graphs with clique number less than k/2, as follows. For every α>0, if ε≤α21350, then if G is an L-critical graph for some k-list-assignment L such that ω(G)<(12−α)k and k is sufficiently large, then G has average degree at least (1+ε)k. This implies that for every α>0, there exists ε>0 such that if G is a graph with ω(G)≤(12−α)mad(G), where mad(G) is the maximum average degree of G, then χℓ(G)≤⌈(1−ε)(mad(G)+1)+εω(G)⌉. It also yields an improvement on the best known upper bound for the chromatic number of Kt-minor free graphs for large t, by a factor of .99982.

    更新日期:2020-01-04
  • The genus of complete 3-uniform hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-08-23
    Yifan Jing; Bojan Mohar

    In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph Kn. In this paper, we investigate the minimum genus embeddings of the complete 3-uniform hypergraphs Kn3. Embeddings of a hypergraph H are defined as the embeddings of its associated Levi graph LH with vertex set V(H)⊔E(H), in which v∈V(H) and e∈E(H) are adjacent if and only if v and e are incident in H. We determine both the orientable and the non-orientable genus of Kn3 when n is even. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of Kn3 is at least 214n2log⁡n(1−o(1)). The construction in the proof may be of independent interest as a design-type problem.

    更新日期:2020-01-04
  • Approximate Moore Graphs are good expanders
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-02
    Michael Dinitz; Michael Schapira; Gal Shahaf

    We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that “sufficiently large” graphs of fixed diameter and degree must be “good” expanders. We prove this statement for various definitions of “sufficiently large” (multiplicative/additive factor from the largest possible size), for different forms of expansion (edge, vertex, and spectral expansion), and for both directed and undirected graphs. A recurring theme is that the lower the diameter of the graph and (more importantly) the larger its size, the better the expansion guarantees. Aside from inherent theoretical interest, our motivation stems from the domain of network design. Both low-diameter networks and expanders are prominent approaches to designing high-performance networks in parallel computing, HPC, datacenter networking, and beyond. Our results establish that these two approaches are, in fact, inextricably intertwined. We leave the reader with many intriguing questions for future research.

    更新日期:2020-01-04
  • Ranking tournaments with no errors I: Structural description
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-08-28
    Xujin Chen; Guoli Ding; Wenan Zang; Qiulan Zhao

    In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments.

    更新日期:2020-01-04
  • N-detachable pairs in 3-connected matroids I: Unveiling X
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-02
    Nick Brettell; Geoff Whittle; Alan Williams

    Let M be a 3-connected matroid, and let N be a 3-connected minor of M. We say that a pair {x1,x2}⊆E(M) is N-detachable if one of the matroids M/x1/x2 or M\x1\x2 is both 3-connected and has an N-minor. This is the first in a series of three papers where we describe the structures that arise when M has no N-detachable pairs. In this paper, we prove that if no N-detachable pair can be found in M, then either M has a 3-separating set, which we call X, with certain strong structural properties, or M has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs.

    更新日期:2020-01-04
  • Factorizing regular graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-05-22
    Carsten Thomassen

    Every 9-regular graph (possibly with multiple edges) with odd edge-connectivity >5 can be edge-decomposed into three 3-factors. If Tutte's 3-flow conjecture is true, it also holds for all 9-regular graphs with odd edge-connectivity 5, but not with odd edge-connectivity 3. It holds for all planar 2-edge-connected 9-regular graphs, an equivalent version of the 4-color theorem for planar graphs. We address the more general question: If G is an r-regular graph, and r=kq where k,q are natural numbers >1, can G be edge-decomposed into k q-factors? If q is even, then the decomposition exists trivially. If k,q are both odd, then we prove that the decomposition exists if G has odd edge-connectivity (size of smallest odd edge-cut) at least 3k−2, which is satisfied if the odd edge-connectivity is at least r−2. If q is odd and k is even, then we must require that G has an even number of vertices just to guarantee that G has a q-factor. If we want a decomposition into q-factors, then we also need the condition that, for any partition of the vertex set of G into two odd parts, there must be at least k edges between the parts. We prove that the edge-decomposition into q-factors is always possible if G has an even number of vertices and the edge-connectivity of G is at least 2k2+k.

    更新日期:2020-01-04
  • The number of Gallai k-colorings of complete graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-31
    Josefran de Oliveira Bastos; Fabrício Siqueira Benevides; Jie Han

    An edge coloring of the n-vertex complete graph, Kn, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for n large and every k with k≤2n/4300, the number of Gallai colorings of Kn that use at most k given colors is ((k2)+on(1))2(n2). Our result is asymptotically best possible and implies that, for those k, almost all Gallai k-colorings use only two colors. However, this is not true for k≥2n/2.

    更新日期:2020-01-04
  • Cycles containing all the odd-degree vertices
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-31
    Kathie Cameron; Carsten Thomassen

    The number of cycles in a graph containing any fixed edge and also containing all vertices of odd degree is odd if and only if all vertices have even degree. If all vertices have even degree this is a theorem of Shunichi Toida. If all vertices have odd degree it is Andrew Thomason's extension of Smith's theorem.

    更新日期:2020-01-04
  • The Kelmans-Seymour conjecture IV: A proof
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-19
    Dawei He; Yan Wang; Xingxing Yu

    A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K5 or K3,3. Wagner proved in 1937 that if a graph other than K5 does not contain any subdivision of K3,3 then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K5 then it is planar or it admits a cut of size at most 4. In this paper, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems.

    更新日期:2020-01-04
  • Finding a path with two labels forbidden in group-labeled graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-16
    Yasushi Kawase; Yusuke Kobayashi; Yutaro Yamaguchi

    The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an s–t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s–t path in a Z3-labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s–t paths in a group-labeled graph or not, and finding s–t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s–t paths, which is the main technical contribution of this paper.

    更新日期:2020-01-04
  • The Kelmans-Seymour conjecture I: Special separations
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-11
    Dawei He; Yan Wang; Xingxing Yu

    Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K5. This conjecture was proved by Ma and Yu for graphs containing K4−, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequent papers to prove the Kelmans-Seymour conjecture.

    更新日期:2020-01-04
  • The Kelmans-Seymour conjecture II: 2-Vertices in K4−
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-11
    Dawei He; Yan Wang; Xingxing Yu

    We use K4− to denote the graph obtained from K4 by removing an edge, and use TK5 to denote a subdivision of K5. Let G be a 5-connected nonplanar graph and {x1,x2,y1,y2}⊆V(G) such that G[{x1,x2,y1,y2}]≅K4− with y1y2∉E(G). Let w1,w2,w3∈N(y2)−{x1,x2} be distinct. We show that G contains a TK5 in which y2 is not a branch vertex, or G−y2 contains K4−, or G has a special 5-separation, or G−{y2v:v∉{w1,w2,w3,x1,x2}} contains TK5. This result will be used to prove the Kelmans-Seymour conjecture that every 5-connected nonplanar graph contains TK5.

    更新日期:2020-01-04
  • The Kelmans-Seymour conjecture III: 3-vertices in K4−
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-09
    Dawei He; Yan Wang; Xingxing Yu

    Let G be a 5-connected nonplanar graph and let x1,x2,y1,y2∈V(G) be distinct, such that G[{x1,x2,y1,y2}]≅K4− and y1y2∉E(G). We show that one of the following holds: G−x1 contains K4−, or G contains a K4− in which x1 is of degree 2, or G contains a TK5 in which x1 is not a branch vertex, or {x2,y1,y2} may be chosen so that for any distinct z0,z1∈N(x1)−{x2,y1,y2}, G−{x1v:v∉{z0,z1,x2,y1,y2}} contains TK5.

    更新日期:2020-01-04
  • On the complex-representable excluded minors for real-representability
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-03
    Rutger Campbell; Jim Geelen

    We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field F1 and a field extension F2, if F1-representability is not equivalent to F2-representability, then each F1-representable matroid is a minor of a F2-representable excluded minor for F1-representability.

    更新日期:2020-01-04
  • Stability and exact Turán numbers for matroids
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-12-03
    Hong Liu; Sammy Luo; Peter Nelson; Kazuhiro Nomoto

    We consider the Turán-type problem of bounding the size of a set M⊆F2n that does not contain a linear copy of a given fixed set N⊆F2k, where n is large compared to k. An Erdős-Stone type theorem [5] in this setting gives a bound that is tight up to a o(2n) error term; our first main result gives a stability version of this theorem, showing that such an M that is close in size to the upper bound in [5] is close to the obvious extremal example in the sense of symmetric difference. Our second result shows that the error term in [5] is exactly controlled by the solution to one of a class of ‘sparse’ extremal problems, and gives some examples where the error term can be eliminated completely to give a sharp upper bound on |M|.

    更新日期:2020-01-04
  • The domination number of plane triangulations
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-11-29
    Simon Špacapan

    We introduce a class of plane graphs called weak near-triangulations, and prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on n>6 vertices has a dominating set of size at most 17n/53. This improves the bound n/3 obtained by Matheson and Tarjan.

    更新日期:2020-01-04
  • On Schelp's problem for three odd long cycles
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-11-25
    Tomasz Łuczak; Zahra Rahimi

    We show that for every η>0 there exists n0 such that for every odd n≥n0 each 3-colouring of edges of a graph G with (4+η)n and minimum degree larger than (7/2+2η)n leads to a monochromatic cycle of length n. This result is, up to η terms, best possible.

    更新日期:2020-01-04
  • A threshold result for loose Hamiltonicity in random regular uniform hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-11-13
    Daniel Altman; Catherine Greenhill; Mikhail Isaev; Reshma Ramadurai

    Let G(n,r,s) denote a uniformly random r-regular s-uniform hypergraph on n vertices, where s is a fixed constant and r=r(n) may grow with n. An ℓ-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely ℓ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When r,s≥3 are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Ruciński and Šileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in G(n,r,s). Finally we prove that for ℓ=2,…,s−1 and for r growing moderately as n→∞, the probability that G(n,r,s) has a ℓ-overlapping Hamilton cycle tends to zero.

    更新日期:2020-01-04
  • On the existence of graphical Frobenius representations and their asymptotic enumeration
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-10-31
    Pablo Spiga

    We give a complete answer to the GFR conjecture, proposed by Conder, Doyle, Tucker and Watkins: “All but finitely many Frobenius groups F=N⋊H with a given complement H have a GFR, with the exception when |H| is odd and N is Abelian but not an elementary 2-group”. Actually, we prove something stronger, we enumerate asymptotically GFRs; we show that, besides the exceptions listed above, as |N| tends to infinity, the proportion of GFRs among all Cayley graphs over N containing F in their automorphism group tends to 1.

    更新日期:2020-01-04
  • The inverse eigenvalue problem of a graph: Multiplicities and minors
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-10-31
    Wayne Barrett; Steve Butler; Shaun M. Fallat; H. Tracy Hall; Leslie Hogben; Jephian C.-H. Lin; Bryan L. Shader; Michael Young

    The inverse eigenvalue problem of a given graph G is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph. Electron. J. Combin., 2017]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonicity, with restrictions on where the new eigenvalues appear. These ideas are applied to solve the inverse eigenvalue problem for all graphs of order five, and to characterize forbidden minors of graphs having at most one multiple eigenvalue.

    更新日期:2020-01-04
  • Cuboids, a class of clutters
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-10-24
    Ahmad Abdi; Gérard Cornuéjols; Natália Guričanová; Dabeen Lee

    The τ=2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special class of clutters. Cuboids are used as means to (a) manifest the geometry behind primal integrality and dual integrality of set covering linear programs, and (b) reveal a geometric rift between these two properties, in turn explaining why primal integrality does not imply dual integrality for set covering linear programs. Along the way, we see that the geometry supports the τ=2 Conjecture. Studying the geometry also leads to over 700 new ideal minimally non-packing clutters over at most 14 elements, a surprising revelation as there was once thought to be only one such clutter.

    更新日期:2020-01-04
  • Ranking tournaments with no errors II: Minimax relation
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-10-24
    Xujin Chen; Guoli Ding; Wenan Zang; Qiulan Zhao

    A tournament T=(V,A) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In the first paper we have given a structural description of all Möbius-free tournaments, and have proved that every CM tournament is Möbius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach.

    更新日期:2020-01-04
  • k-regular subgraphs near the k-core threshold of a random graph
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-10-14
    Dieter Mitsche; Michael Molloy; Paweł Prałat

    We prove that Gn,p=c/n w.h.p. has a k-regular subgraph if c is at least e−Θ(k) above the threshold for the appearance of a subgraph with minimum degree at least k; i.e. a non-empty k-core. In particular, this pins down the threshold for the appearance of a k-regular subgraph to a window of size e−Θ(k).

    更新日期:2020-01-04
  • Matroid fragility and relaxations of circuit hyperplanes
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-26
    Jim Geelen; Florian Hoersch

    We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let F be a finite field. The first conjecture states that: the branch-width of any F-representable N-fragile matroid is bounded by a function depending only upon F and N. The second conjecture states that: if a matroid M2 is obtained from a matroid M1 by relaxing a circuit-hyperplane and both M1 and M2 are F-representable, then the branch-width of M1 is bounded by a function depending only upon F. Our main result is that the second conjecture implies the first.

    更新日期:2020-01-04
  • A large number of m-coloured complete infinite subgraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-24
    António Girão

    Given an edge colouring of a graph with a set of m colours, we say that the graph is m-coloured if each of the m colours is used. For an m-colouring Δ of N(2), the complete graph on N, we denote by FΔ the set all values γ for which there exists an infinite subset X⊂N such that X(2) is γ-coloured. Properties of this set were first studied by Erickson in 1994. Here, we are interested in estimating the minimum size of FΔ over all m-colourings Δ of N(2). Indeed, we shall prove the following result. There exists an absolute constant α>0 such that for any positive integer m≠{(n2)+1,(n2)+2:n≥2}, |FΔ|≥(1+α)2m, for any m-colouring Δ of N(2). This proves a conjecture of Narayanan. We remark the result is tight up to the value of α.

    更新日期:2020-01-04
  • Simple k-planar graphs are simple (k + 1)-quasiplanar
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-09
    Patrizio Angelini; Michael A. Bekos; Franz J. Brandenburg; Giordano Da Lozzo; Giuseppe Di Battista; Walter Didimo; Michael Hoffmann; Giuseppe Liotta; Fabrizio Montecchiani; Ignaz Rutter; Csaba D. Tóth

    A simple topological graph is k-quasiplanar (k≥2) if it contains no k pairwise crossing edges, and k-planar if no edge is crossed more than k times. In this paper, we explore the relationship between k-planarity and k-quasiplanarity to show that, for k≥2, every k-planar simple topological graph can be transformed into a (k+1)-quasiplanar simple topological graph.

    更新日期:2020-01-04
  • Towards the linear arboricity conjecture
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-09
    Asaf Ferber; Jacob Fox; Vishesh Jain

    The linear arboricity of a graph G, denoted by la(G), is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in G whose union covers all the edges of G. A famous conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that la(G)≤⌈(Δ(G)+1)/2⌉, where Δ(G) denotes the maximum degree of G. This conjectured upper bound would be best possible, as is easily seen by taking G to be a regular graph. In this paper, we show that for every graph G, la(G)≤Δ2+O(Δ2/3−α) for some α>0, thereby improving the previously best known bound due to Alon and Spencer from 1992. For graphs which are sufficiently good spectral expanders, we give even better bounds. Our proofs of these results further give probabilistic polynomial time algorithms for finding such decompositions into linear forests.

    更新日期:2020-01-04
  • Induced subgraphs of graphs with large chromatic number. VII. Gyárfás' complementation conjecture
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-09-05
    Alex Scott; Paul Seymour

    A class of graphs is χ-bounded if there is a function f such that χ(G)≤f(ω(G)) for every induced subgraph G of every graph in the class, where χ,ω denote the chromatic number and clique number of G respectively. In 1987, Gyárfás conjectured that for every c, if C is a class of graphs such that χ(G)≤ω(G)+c for every induced subgraph G of every graph in the class, then the class of complements of members of C is χ-bounded. We prove this conjecture. Indeed, more generally, a class of graphs is χ-bounded if it has the property that no graph in the class has c+1 odd holes, pairwise disjoint and with no edges between them. The main tool is a lemma that if C is a shortest odd hole in a graph, and X is the set of vertices with at least five neighbours in V(C), then there is a three-vertex set that dominates X.

    更新日期:2020-01-04
  • The (theta, wheel)-free graphs Part II: Structure theorem
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-08-19
    Marko Radovanović; Nicolas Trotignon; Kristina Vušković

    A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced subgraph isomorphic to a theta or a wheel, i.e. the class of (theta, wheel)-free graphs. The decomposition theorem uses clique cutsets and 2-joins. Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graph classes. A 2-join is an edge cutset that appeared in decomposition theorems of several complex classes, such as perfect graphs, even-hole-free graphs and others. In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations (which are much harder to use in algorithms). This is a first example of a decomposition theorem that uses just the combination of clique cutsets and 2-joins. This has several consequences. First, we can easily transform our decomposition theorem into a complete structure theorem for (theta, wheel)-free graphs, i.e. we show how every (theta, wheel)-free graph can be built starting from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are (theta, wheel)-free. Such structure theorems are very rare for hereditary graph classes, only a few examples are known. Secondly, we obtain an O(n4m)-time decomposition based recognition algorithm for (theta, wheel)-free graphs. Finally, in Parts III and IV of this series, we give further applications of our decomposition theorem.

    更新日期:2020-01-04
  • The (theta, wheel)-free graphs Part III: Cliques, stable sets and coloring
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-07-17
    Marko Radovanović; Nicolas Trotignon; Kristina Vušković

    A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a vertex that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins, and consequently obtain a polynomial time recognition algorithm for the class. In this paper we further use this decomposition theorem to obtain polynomial time algorithms for maximum weight clique, maximum weight stable set and coloring problems. We also show that for a graph G in the class, if its maximum clique size is ω, then its chromatic number is bounded by max⁡{ω,3}, and that the class is 3-clique-colorable.

    更新日期:2020-01-04
  • The (theta, wheel)-free graphs Part I: Only-prism and only-pyramid graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2018-02-03
    Emilie Diot; Marko Radovanović; Nicolas Trotignon; Kristina Vušković

    Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence is a polynomial time recognition algorithms for these two classes. In Part II of this series we generalize these results to graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV, using the obtained structure, we solve several optimization problems for these graphs.

    更新日期:2020-01-04
  • New code upper bounds for the folded n-cube
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-26
    Lihang Hou; Bo Hou; Suogang Gao; Wei-Hsuan Yu

    Let □n denote the folded n-cube and let A(□n,d) denote the maximum size of a code in □n with minimum distance at least d. We give an upper bound on A(□n,d) based on block-diagonalizing the Terwilliger algebra of □n and on semidefinite programming. The technique of this paper is an extension of the approach taken by A. Schrijver [11] on the study of upper bounds for binary codes.

    更新日期:2020-01-04
  • Erdős-Ko-Rado theorems for set partitions with certain block size
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-12
    Cheng Yeaw Ku; Kok Bin Wong

    In this paper, we prove Erdős-Ko-Rado type results for (a) family of set partitions where the size of each block is a multiple of k; and (b) family of set partitions with minimum block size k.

    更新日期:2020-01-04
  • State transfer in strongly regular graphs with an edge perturbation
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-12
    Chris Godsil; Krystal Guo; Mark Kempton; Gabor Lippner; Florentin Münch

    Quantum walks, an important tool in quantum computing, have been very successfully investigated using techniques in algebraic graph theory. We are motivated by the study of state transfer in continuous-time quantum walks, which is understood to be a rare and interesting phenomenon. We consider a perturbation on an edge uv of a graph where we add a weight β to the edge and a loop of weight γ to each of u and v. We characterize when this perturbation results in strongly cospectral vertices u and v. Applying this to strongly regular graphs, we give infinite families of strongly regular graphs where some perturbation results in perfect state transfer. Further, we show that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer. We also show for any strongly regular graph X and edge e∈E(X), that ϕ(X\e) does not depend on the choice of e.

    更新日期:2020-01-04
  • The generating function of planar Eulerian orientations
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-16
    Mireille Bousquet-Mélou; Andrew Elvey Price

    The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model. We solve both problems – namely the enumeration of planar Eulerian orientations and of 4-valent planar Eulerian orientations – by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, μn/(nlog⁡n)2, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order 2. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model. Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4-valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially.

    更新日期:2020-01-04
  • Pattern groups and a poset based Hopf monoid
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-27
    Farid Aliniaeifard; Nathaniel Thiem

    The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

    更新日期:2020-01-04
  • Asymptotic enumeration of compacted binary trees of bounded right height
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-30
    Antoine Genitrini; Bernhard Gittenberger; Manuel Kauers; Michael Wallner

    A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf. We solve the asymptotic counting problem for this class as well as a closely related, further simplified class. For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations. Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity.

    更新日期:2020-01-04
  • Winding of simple walks on the square lattice
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-31
    Timothy Budd

    A method is described to count simple diagonal walks on Z2 with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the generating functions of which are encoded in a commuting set of Hilbert space operators. The general enumeration problem is then solved by obtaining an explicit eigenvalue decomposition of these operators involving elliptic functions. By further restricting the intermediate winding angles of the walks to some open interval, the method can be used to count various classes of walks restricted to cones in Z2 of opening angles that are integer multiples of π/4. We present three applications of this main result. First we find an explicit generating function for the walks in such cones that start and end at the origin. In the particular case of a cone of angle 3π/4 these walks are directly related to Gessel's walks in the quadrant, and we provide a new proof of their enumeration. Next we study the distribution of the winding angle of a simple random walk on Z2 around a point in the close vicinity of its starting point, for which we identify discrete analogues of the known hyperbolic secant laws and a probabilistic interpretation of the Jacobi elliptic functions. Finally we relate the spectrum of one of the Hilbert space operators to the enumeration of closed loops in Z2 with fixed winding number around the origin.

    更新日期:2020-01-04
  • Vertex-isoperimetric stability in the hypercube
    J. Comb. Theory A (IF 0.958) Pub Date : 2019-12-31
    Michał Przykucki; Alexander Roberts

    Harper's Theorem states that, in a hypercube, among all sets of a given fixed size the Hamming balls have minimal closed neighbourhoods. In this paper we prove a stability-like result for Harper's Theorem: if the closed neighbourhood of a set is close to minimal in the hypercube, then the set must be very close to a Hamming ball around some vertex.

    更新日期:2020-01-04
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