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On the spectral gap and the automorphism group of distance-regular graphs J. Comb. Theory B (IF 1.306) Pub Date : 2021-02-26 Bohdan Kivva
We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group. The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree
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Connectivity and choosability of graphs with no Kt minor J. Comb. Theory B (IF 1.306) Pub Date : 2021-02-18 Sergey Norin; Luke Postle
In 1943, Hadwiger conjectured that every graph with no Kt+1 minor is t-colorable for every t≥0. While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980 s, Kostochka and Thomason independently proved that every graph with no Kt minor has average degree O(tlogt) and thus is O(tlogt)-list-colorable. Recently, the authors and Song proved
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A solution of Li-Xia's problem on s-arc-transitive solvable Cayley graphs J. Comb. Theory B (IF 1.306) Pub Date : 2021-02-15 Jin-Xin Zhou
This paper gives a solution of Problem 1.8 in [16]. As a corollary, it is shown that every connected non-bipartite Cayley graph on a solvable group of valency at least three is at most 2-arc-transitive.
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Exponentially many 3-colorings of planar triangle-free graphs with no short separating cycles J. Comb. Theory B (IF 1.306) Pub Date : 2021-02-08 Carsten Thomassen
The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2n/17700000. On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is <215n/log2(n).
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Total weight choosability of graphs: Towards the 1-2-3-conjecture J. Comb. Theory B (IF 1.306) Pub Date : 2021-02-03 Lu Cao
Let G=(V,E) be a graph. A proper total weighting of G is a mapping w:V∪E⟶R such that the following sum for each v∈V:w(v)+∑e∈E(v)w(e) gives a proper vertex colouring of G. For any a,b∈N+, we say that G is total weight (a,b)-choosable if for any {Sv:v∈V}⊂[R]a and {Se:v∈E}⊂[R]b, there exists a proper total weighting w of G such that w(v)∈Sv for v∈V and w(e)∈Se for e∈E. A strengthening of the 1-2-3 Conjecture
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Some extremal results on 4-cycles J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-29 Jialin He; Jie Ma; Tianchi Yang
We present two extremal results on 4-cycles. Let q be a large even integer. First we prove that every (q2+q+1)-vertex C4-free graph with more than 12q(q+1)2−0.2q edges must be a spanning subgraph of a unique polarity graph. This implies a stability refinement of a special case of the seminal work of Füredi on the extremal number of C4. Second we prove that every (q2+q+1)-vertex graph with 12q(q+1)2+1
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Obstructions for bounded shrub-depth and rank-depth J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-25 O-joung Kwon; Rose McCarty; Sang-il Oum; Paul Wollan
Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný et al. (2016) [11]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long
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Acyclic orientation polynomials and the sink theorem for chromatic symmetric functions J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-22 Byung-Hak Hwang; Woo-Seok Jung; Kang-Ju Lee; Jaeseong Oh; Sang-Hoon Yu
We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at −1 up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop “acyclic orientation” analogues of theorems concerning the
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The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-20 Leonardo Martínez-Sandoval; Arnau Padrol
The convex dimension of a k-uniform hypergraph is the smallest dimension d for which there is an injective mapping of its vertices into Rd such that the set of k-barycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete k-uniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We
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A Cantor-Bernstein-type theorem for spanning trees in infinite graphs J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-19 Joshua Erde; J. Pascal Gollin; Atilla Joó; Paul Knappe; Max Pitz
We show that if a graph admits a packing and a covering both consisting of λ many spanning trees, where λ is some infinite cardinal, then the graph also admits a decomposition into λ many spanning trees. For finite λ the analogous question remains open, however, a slightly weaker statement is proved.
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A Harary-Sachs theorem for hypergraphs J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-14 Gregory J. Clark; Joshua N. Cooper
We generalize the Harary-Sachs theorem to k-uniform hypergraphs: the codegree-d coefficient of the characteristic polynomial of a uniform hypergraph H can be expressed as a weighted sum of subgraph counts over certain multi-hypergraphs with d edges. We include a detailed description of the aforementioned multi-hypergraphs and a formula for their corresponding weights.
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Long cycles, heavy cycles and cycle decompositions in digraphs J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-11 Charlotte Knierim; Maxime Larcher; Anders Martinsson; Andreas Noever
Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most ⌊(n−1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1], Dean (1986) [7], and Bollobás and Scott (1996) [2] it was analogously conjectured that every directed Eulerian graph can be decomposed
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On the rational Turán exponents conjecture J. Comb. Theory B (IF 1.306) Pub Date : 2021-01-12 Dong Yeap Kang; Jaehoon Kim; Hong Liu
The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n,F)=Θ(nr). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 0,1,75,2, and the numbers
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Regular Cayley maps for dihedral groups J. Comb. Theory B (IF 1.306) Pub Date : 2020-12-29 István Kovács; Young Soo Kwon
An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut∘M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut∘M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified
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The feasible region of hypergraphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-12-24 Xizhi Liu; Dhruv Mubayi
Let F be a family of r-uniform hypergraphs. The feasible region Ω(F) of F is the set of points (x,y) in the unit square such that there exists a sequence of F-free r-uniform hypergraphs whose shadow density approaches x and whose edge density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x,y)∈Ω(F) is the Turán density π(F), and
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Kr-factors in graphs with low independence number J. Comb. Theory B (IF 1.306) Pub Date : 2020-12-24 Charlotte Knierim; Pascal Su
A classical result by Hajnal and Szemerédi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a Kr-factor. Namely, any graph on n vertices, with minimum degree δ(G)≥(1−1r)n and r dividing n has a Kr-factor. This result is tight but the extremal examples are unique in that they all have a large independent set which is the bottleneck. Nenadov and Pehova
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A Menger-like property of tree-cut width J. Comb. Theory B (IF 1.306) Pub Date : 2020-12-22 Archontia C. Giannopoulou; O-joung Kwon; Jean-Florent Raymond; Dimitrios M. Thilikos
In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan (2015) as a possible edge-version of tree decompositions. We show
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Tetravalent half-arc-transitive graphs with unbounded nonabelian vertex stabilizers J. Comb. Theory B (IF 1.306) Pub Date : 2020-12-16 Binzhou Xia
Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian
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Graph functionality J. Comb. Theory B (IF 1.306) Pub Date : 2020-11-17 Bogdan Alecu; Aistis Atminas; Vadim Lozin
In the present paper, we introduce the notion of graph functionality, which generalises simultaneously several other graph parameters, such as degeneracy or clique-width, in the sense that bounded degeneracy or bounded clique-width imply bounded functionality. Moreover, we show that this generalisation is proper by revealing classes of graphs of unbounded degeneracy and clique-width, where functionality
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Reconfiguring colorings of graphs with bounded maximum average degree J. Comb. Theory B (IF 1.306) Pub Date : 2020-11-12 Carl Feghali
The reconfiguration graph Rk(G) for the k-colorings of a graph G has as vertex set the set of all possible k-colorings of G and two colorings are adjacent if they differ in the color of exactly one vertex of G. Let d,k≥1 be integers such that k≥d+1. We prove that for every ϵ>0 and every graph G with n vertices and maximum average degree d−ϵ, Rk(G) has diameter O(n(logn)d−1). This significantly strengthens
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Sparse hypergraphs: New bounds and constructions J. Comb. Theory B (IF 1.306) Pub Date : 2020-11-05 Gennian Ge; Chong Shangguan
Let fr(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v+1 vertices. The study of fr(n,v,e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known. Conjecture: nk−o(1)k≥2 and e≥3 as n→∞. For r=3,e=3,k=2, the bound n2−o(1)
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Dismantlability, connectedness, and mixing in relational structures J. Comb. Theory B (IF 1.306) Pub Date : 2020-10-22 Raimundo Briceño; Andrei Bulatov; Víctor Dalmau; Benoît Larose
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have been shown to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved—like, for example, dismantlability—and
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Stability of graph pairs J. Comb. Theory B (IF 1.306) Pub Date : 2020-10-27 Yan-Li Qin; Binzhou Xia; Jin-Xin Zhou; Sanming Zhou
We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs (Γ,Σ) is stable if Aut(Γ×Σ)≅Aut(Γ)×Aut(Σ) and unstable otherwise, where Γ×Σ is the direct product of Γ and Σ. An unstable graph pair (Γ,Σ) is said to be a nontrivially unstable graph pair if Γ and Σ are connected coprime graphs, at least
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3-List-coloring graphs of girth at least five on surfaces J. Comb. Theory B (IF 1.306) Pub Date : 2020-10-13 Luke Postle
Grötzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any
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A unified view of inequalities for distance-regular graphs, part I J. Comb. Theory B (IF 1.306) Pub Date : 2020-10-05 Arnold Neumaier; Safet Penjić
In this paper, we introduce the language of a configuration and of t-point counts for distance-regular graphs (DRGs). Every t-point count can be written as a sum of (t−1)-point counts. This leads to a system of linear equations and inequalities for the t-point counts in terms of the intersection numbers, i.e., a linear constraint satisfaction problem (CSP). This language is a very useful tool for a
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On some topological and combinatorial lower bounds on the chromatic number of Kneser type hypergraphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-10-01 Soheil Azarpendar; Amir Jafari
In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers n,k,r,s with n≥rk and 2≤s≤r, the r-uniform general Kneser hypergraph KGsr(n,k), has all k-subsets of {1,…,n} as the vertex set and all multi-sets {A1,…,Ar} of k-subsets with s-wise empty intersections as the edge set. The case r=s=2, was considered by Kneser
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A counterexample to prism-hamiltonicity of 3-connected planar graphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-29 Simon Špacapan
The prism over a graph G is the Cartesian product of G with the complete graph K2. A graph G is hamiltonian if there exists a spanning cycle in G, and G is prism-hamiltonian if the prism over G is hamiltonian. Rosenfeld and Barnette (1973) [15] conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.
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The Grid Theorem for vertex-minors J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-28 Jim Geelen; O-joung Kwon; Rose McCarty; Paul Wollan
We prove that, for each circle graph H, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to H.
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The templates for some classes of quaternary matroids J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-28 Kevin Grace
Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the eventual extremal functions for these classes. One of the main tools for obtaining these results is the notion of
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Subdivisions of digraphs in tournaments J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-25 António Girão; Kamil Popielarz; Richard Snyder
We show that for every positive integer k, any tournament with minimum out-degree at least (2+o(1))k2 contains a subdivision of the complete directed graph on k vertices, where each path of the subdivision has length at most 3. This result is best possible on the minimum out-degree condition (up to a multiplicative factor of 8), and it is tight with respect to the length of the paths. It may be viewed
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Towards tight(er) bounds for the Excluded Grid Theorem J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-25 Julia Chuzhoy; Zihan Tan
We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function f:Z+→Z+, such that for every integer g>0, every graph of treewidth at least f(g) contains the (g×g)-grid as a minor. For every integer g>0, let f(g) be the smallest value for which the theorem
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Global rigidity of periodic graphs under fixed-lattice representations J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-24 Viktória E. Kaszanitzky; Bernd Schulze; Shin-ichi Tanigawa
In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of
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Relatively small counterexamples to Hedetniemi's conjecture J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-23 Xuding Zhu
Hedetniemi conjectured in 1966 that χ(G×H)=min{χ(G),χ(H)} for all graphs G and H. Here G×H is the graph with vertex set V(G)×V(H) defined by putting (x,y) and (x′,y′) adjacent if and only if xx′∈E(G) and yy′∈E(H). This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let p be the minimum number of vertices in a graph of odd girth 7 and fractional
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Asymptotic density of graphs excluding disconnected minors J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-23 Rohan Kapadia; Sergey Norin; Yingjie Qian
For a graph H, letc∞(H)=limn→∞max|E(G)|n, where the maximum is taken over all graphs G on n vertices not containing H as a minor. Thus c∞(H) is the asymptotic maximum density of graphs not containing H as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on c∞(H) for disconnected graphs H. In particular, we determine c∞(H) whenever H is a union of cycles. Finally, we
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Corrigendum to “Cycle decompositions of Kn and Kn − I” [J. Combin. Theory Ser. B 81 (1) (2001) 77–99] J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-23 Brian Alspach; Heather Jordon
There is error in the proof of Theorem 1.1, Subcase 2.1 of [J. Combin. Theory Ser. B 81 (1) (2001) 77–99]. Here, we fix that error.
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Additive non-approximability of chromatic number in proper minor-closed classes J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-23 Zdeněk Dvořák; Ken-ichi Kawarabayashi
Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally
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Ramsey numbers of path-matchings, covering designs, and 1-cores J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-16 Louis DeBiasio, András Gyárfás, Gábor N. Sárközy
A path-matching of order p is a vertex disjoint union of nontrivial paths spanning p vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers r,p1,…,pr, define RPM(p1,…,pr) to be the smallest integer n such that in any r-coloring of the edges of Kn there exists
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Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-09-14 Zdeněk Dvořák; Daniel Král'; Robin Thomas
Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that∑fface ofG(|f|−4)≤κ(g+t+c−1) for a fixed constant κ, thus generalizing and strengthening several known results. As a corollary, we prove that every triangle-free graph G embedded in a surface of genus g contains a set of O(g) vertices such
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Minimum degree conditions for monochromatic cycle partitioning J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-27 Dániel Korándi, Richard Lang, Shoham Letzter, Alexey Pokrovskiy
A classical result of Erdős, Gyárfás and Pyber states that any r-edge-coloured complete graph has a partition into O(r2logr) monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant c such that any r-edge-coloured graph on n vertices with minimum degree at least n/2+c⋅rlogn has a partition into O(r2) monochromatic
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A strengthening of Erdős-Gallai Theorem and proof of Woodall's conjecture J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-24 Binlong Li, Bo Ning
For a 2-connected graph G on n vertices and two vertices x,y∈V(G), we prove that there is an (x,y)-path of length at least k, if there are at least n−12 vertices in V(G)\{x,y} of degree at least k. This strengthens a celebrated theorem due to Erdős and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with n vertices contains a cycle of length at least 2k, if
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Gaps in the cycle spectrum of 3-connected cubic planar graphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-19 Martin Merker
We prove that, for every natural number k, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in [k,2k+9]. We also show that this bound is close to being optimal by constructing, for every even k≥4, an infinite family of 3-connected cubic planar graphs that contain no cycle whose length is in [k,2k+1].
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The interval number of a planar graph is at most three J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-12 Guillaume Guégan, Kolja Knauer, Jonathan Rollin, Torsten Ueckerdt
The interval number of a graph G is the minimum k such that one can assign to each vertex of G a union of k intervals on the real line, such that G is the intersection graph of these sets, i.e., two vertices are adjacent in G if and only if the corresponding sets of intervals have non-empty intersection. Scheinerman and West (1983) [14] proved that the interval number of any planar graph is at most
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N-detachable pairs in 3-connected matroids II: Life in X J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-10 Nick Brettell; Geoff Whittle; Alan Williams
Let M be a 3-connected matroid, and let N be a 3-connected minor of M. 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 second in a series of three papers where we describe the structures that arise when it is not possible to find an N-detachable pair in M. In the first paper in the series, we showed that, under mild assumptions
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Tangles and the Stone-Čech compactification of infinite graphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-07 Jan Kurkofka, Max Pitz
We show that the tangle space of a graph, which compactifies it, is a quotient of its Stone-Čech remainder obtained by contracting the connected components.
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Most Laplacian eigenvalues of a tree are small J. Comb. Theory B (IF 1.306) Pub Date : 2020-08-06 David P. Jacobs, Elismar R. Oliveira, Vilmar Trevisan
We show that the number of Laplacian eigenvalues less than the average degree 2−2n of a tree having n vertices is at least ⌈n2⌉.
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On the average degree of edge chromatic critical graphs II J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-31 Yan Cao, Guantao Chen
In the study of graph edge coloring for simple graphs, a graph G is called Δ-critical if Δ(G)=Δ, χ′(G)=Δ(G)+1 and χ′(H)<χ′(G) for every proper subgraph H of G. In this paper, we prove a new adjacency result of critical graphs which allows us to control the degree of vertices with distance four. Combining this result with a previous theorem proved by the authors, we show that for every ϵ>0, if G is
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A unified existence theorem for normal spanning trees J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-28 Max Pitz
We show that a graph G has a normal spanning tree if and only if its vertex set is the union of countably many sets each separated from any subdivided infinite clique in G by a finite set of vertices. This proves a conjecture by Brochet and Diestel from 1994, giving a common strengthening of two classical normal spanning tree criterions due to Jung and Halin. Moreover, our method gives a new, algorithmic
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Super-pancyclic hypergraphs and bipartite graphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-16 Alexandr Kostochka, Ruth Luo, Dara Zirlin
We find Dirac-type sufficient conditions for a hypergraph H with few edges to be hamiltonian. We also show that these conditions guarantee that H is super-pancyclic, i.e., for each A⊆V(H) with |A|≥3, H contains a Berge cycle with vertex set A. We mostly use the language of bipartite graphs, because every bipartite graph is the incidence graph of a multihypergraph. In particular, we extend some results
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The pseudoforest analogue for the Strong Nine Dragon Tree Conjecture is true J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-14 Logan Grout, Benjamin Moore
We prove that for any positive integers k and d, if a graph G has maximum average degree at most 2k+2dd+k+1, then G decomposes into k+1 pseudoforests C1,…,Ck+1 so that for at least one of the pseudoforests, each connected component has at most d edges.
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The (theta, wheel)-free graphs Part IV: Induced paths and cycles J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-07 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 internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node 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
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Maker-breaker percolation games II: Escaping to infinity J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-02 A. Nicholas Day; Victor Falgas–Ravry
Let Λ be an infinite connected graph, and let v0 be a vertex of Λ. We consider the following positional game. Two players, Maker and Breaker, play in alternating turns. Initially all edges of Λ are marked as unsafe. On each of her turns, Maker marks p unsafe edges as safe, while on each of his turns Breaker takes q unsafe edges and deletes them from the graph. Breaker wins if at any time in the game
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Three-coloring triangle-free graphs on surfaces III. Graphs of girth five J. Comb. Theory B (IF 1.306) Pub Date : 2020-07-02 Zdeněk Dvořák, Daniel Král', Robin Thomas
We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.
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The multicolour size-Ramsey number of powers of paths J. Comb. Theory B (IF 1.306) Pub Date : 2020-06-27 Jie Han, Matthew Jenssen, Yoshiharu Kohayakawa, Guilherme Oliveira Mota, Barnaby Roberts
Given a positive integer s, a graph G is s-Ramsey for a graph H, denoted G→(H)s, if every s-colouring of the edges of G contains a monochromatic copy of H. The s-colour size-Ramsey number rˆs(H) of a graph H is defined to be rˆs(H)=min{|E(G)|:G→(H)s}. We prove that, for all positive integers k and s, we have rˆs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn.
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An asymptotic resolution of a problem of Plesník J. Comb. Theory B (IF 1.306) Pub Date : 2020-06-22 Stijn Cambie
Fix d≥3. We show the existence of a constant c>0 such that any graph of diameter at most d has average distance at most d−cd3/2n, where n is the number of vertices. Moreover, we exhibit graphs certifying sharpness of this bound up to the choice of c. This constitutes an asymptotic solution to a longstanding open problem of Plesník. Furthermore we solve the problem exactly for digraphs if the order
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Contact graphs of ball packings J. Comb. Theory B (IF 1.306) Pub Date : 2020-06-17 Alexey Glazyrin
A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in R3 is not greater than 13.92. We also find new upper bounds for the average degree of contact graphs in R4 and R5.
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Circular flows via extended Tutte orientations J. Comb. Theory B (IF 1.306) Pub Date : 2020-06-16 Jiaao Li, Yezhou Wu, Cun-Quan Zhang
This paper consists of two major parts. In the first part, the relations between Tutte orientations and circular flows are explored. Tutte orientation (modulo orientation) was first observed by Tutte for the study of 3-flow problem, and later extended by Jaeger for circular (2+1/p)-flows. In this paper, it is extended for circular λ-flows for all rational numbers λ. This theorem is one of the key tools
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Edge-transitive bi-Cayley graphs J. Comb. Theory B (IF 1.306) Pub Date : 2020-06-09 Marston Conder, Jin-Xin Zhou, Yan-Quan Feng, Mi-Mi Zhang
A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism
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Planar graphs that need four pages J. Comb. Theory B (IF 1.306) Pub Date : 2020-06-04 Mihalis Yannakakis
We show that there are planar graphs that require four pages in any book embedding.
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Completion and deficiency problems J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-29 Rajko Nenadov, Benny Sudakov, Adam Zsolt Wagner
Given a partial Steiner triple system (STS) of order n, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order n with at most r≤εn2 triples, it can always be embedded into a complete STS of order n+O(r), which is asymptotically optimal
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Generalized Turán problems for even cycles J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-27 Dániel Gerbner, Ervin Győri, Abhishek Methuku, Máté Vizer
Given a graph H and a set of graphs F, let ex(n,H,F) denote the maximum possible number of copies of H in an F-free graph on n vertices. We investigate the function ex(n,H,F), when H and members of F are cycles. Let Ck denote the cycle of length k and let Ck={C3,C4,…,Ck}. We highlight the main results below. (i) We show that ex(n,C2l,C2k)=Θ(nl) for any l,k≥2. Moreover, in some cases we determine it