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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

    更新日期:2020-09-23
  • 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

    更新日期:2020-09-23
  • 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.

    更新日期:2020-09-23
  • 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

    更新日期:2020-09-23
  • 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

    更新日期:2020-09-16
  • 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

    更新日期:2020-09-14
  • 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(r2log⁡r) 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⋅rlog⁡n has a partition into O(r2) monochromatic

    更新日期:2020-08-27
  • 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

    更新日期:2020-08-24
  • 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].

    更新日期:2020-08-19
  • 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

    更新日期:2020-08-12
  • 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

    更新日期:2020-08-11
  • 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.

    更新日期:2020-08-07
  • 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⌉.

    更新日期:2020-08-06
  • 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

    更新日期:2020-07-31
  • 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

    更新日期:2020-07-28
  • 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

    更新日期:2020-07-16
  • 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.

    更新日期:2020-07-14
  • 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

    更新日期:2020-07-07
  • 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

    更新日期:2020-07-02
  • 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.

    更新日期:2020-07-02
  • 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.

    更新日期:2020-06-27
  • 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

    更新日期:2020-06-22
  • 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.

    更新日期:2020-06-17
  • 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

    更新日期:2020-06-16
  • 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

    更新日期:2020-06-09
  • 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.

    更新日期:2020-06-04
  • 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

    更新日期:2020-05-29
  • 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

    更新日期:2020-05-27
  • Global rigidity of direction-length frameworks
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-25
    Katie Clinch, Bill Jackson, Peter Keevash

    A 2-dimensional direction-length framework is a collection of points in the plane which are linked by pairwise constraints that fix the direction or length of the line segments joining certain pairs of points. We represent it as a pair (G,p), where G=(V;D,L) is a ‘mixed’ graph and p:V→R2 is a point configuration for V. It is globally rigid if every direction-length framework (G,q) which satisfies the

    更新日期:2020-05-25
  • Stability for vertex isoperimetry in the cube
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-22
    Peter Keevash, Eoin Long

    We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local stability result for ball-like sets that gives a sharp estimate for the vertex boundary in terms of the distance from a ball, and so our stability result is essentially

    更新日期:2020-05-22
  • Erdős-Pósa property of chordless cycles and its applications
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-20
    Eun Jung Kim, O-joung Kwon

    A chordless cycle, or equivalently a hole, in a graph G is an induced subgraph of G which is a cycle of length at least 4. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k+1 vertex-disjoint chordless cycles, or c1k2log⁡k+c2

    更新日期:2020-05-20
  • Decompositions of complete multigraphs into stars of varying sizes
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-19
    Rosalind A. Cameron, Daniel Horsley

    In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size m exists whenever some obvious necessary conditions hold. In 1992 Lonc gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple) complete graph into stars of sizes m1,…,mt. We show that the general problem of when a complete multigraph admits a decomposition into stars

    更新日期:2020-05-19
  • Homomorphisms of partial t-trees and edge-colorings of partial 3-trees
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-13
    Meirun Chen, Reza Naserasr

    A reformulation of the four color theorem is to say that K4 is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization

    更新日期:2020-05-13
  • Relating the cut distance and the weak* topology for graphons
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-07
    Martin Doležal; Jan Grebík; Jan Hladký; Israel Rocha; Václav Rozhoň

    The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach the cut norm topology via the weak* topology (when considering a predual of L1-functions). We prove that a sequence W1,W2,W3,… of graphons converges in the cut distance if and only if we have equality of the sets of weak* accumulation points and of weak* limit points of all sequences of graphons W1′

    更新日期:2020-05-07
  • Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-05
    Zdeněk Dvořák; Daniel Král'; Robin Thomas

    We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified

    更新日期:2020-05-05
  • The structure of hypergraphs without long Berge cycles
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-05
    Ervin Győri; Nathan Lemons; Nika Salia; Oscar Zamora

    We study the structure of r-uniform hypergraphs containing no Berge cycles of length at least k for k≤r, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such hypergraphs, giving an affirmative answer to the conjectured value when k=r and giving a simple solution to a recent result of Kostochka-Luo when k

    更新日期:2020-05-05
  • A localization method in Hamiltonian graph theory
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-05
    Armen S. Asratian; Jonas B. Granholm; Nikolay K. Khachatryan

    The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea

    更新日期:2020-05-05
  • Flows on flow-admissible signed graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-05
    Matt DeVos; Jiaao Li; You Lu; Rong Luo; Cun-Quan Zhang; Zhang Zhang

    In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero 6-flow. Bouchet himself proved that such signed graphs admit nowhere-zero 216-flows and Zýka further proved that such signed graphs admit nowhere-zero 30-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero 11-flow.

    更新日期:2020-05-05
  • On the average degree of edge chromatic critical graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-05-05
    Yan Cao; Guantao Chen

    Let G be a simple graph, and let χ′(G) and Δ(G) denote the chromatic index and the maximum degree of G, respectively. A graph G is a critical class two graph if χ′(G)=Δ(G)+1 and χ′(H)≤Δ(G) for every proper subgraph H of G. Let d‾(G) denote the average degree of G, i.e., d‾(G)=2|E(G)|/|V(G)|. Vizing in 1968 conjectured that d‾(G)≥Δ(G)−1+3/n if G is a critical class two graph of order n. In this paper

    更新日期:2020-05-05
  • Maximising the number of cycles in graphs with forbidden subgraphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-04-16
    Natasha Morrison; Alexander Roberts; Alex Scott

    Fix k≥2 and let H be a graph with χ(H)=k+1 containing a critical edge. We show that for sufficiently large n, the unique n-vertex H-free graph containing the maximum number of cycles is Tk(n). This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian [4].

    更新日期:2020-04-21
  • Disproving the normal graph conjecture
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-04-16
    Ararat Harutyunyan; Lucas Pastor; Stéphan Thomassé

    A graph G is called normal if there exist two coverings, C and S of its vertex set such that every member of C induces a clique in G, every member of S induces an independent set in G and C∩S≠∅ for every C∈C and S∈S. It has been conjectured by De Simone and Körner in 1999 that a graph G is normal if G does not contain C5, C7 and C7‾ as an induced subgraph. We disprove this conjecture.

    更新日期:2020-04-21
  • The 1–2–3 Conjecture almost holds for regular graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-25
    Jakub Przybyło

    The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set {1,2,3,4,5}. We show that for regular graphs it is sufficient to use weights 1, 2, 3, 4. Moreover, we prove the conjecture to hold for every

    更新日期:2020-04-21
  • Hedetniemi's conjecture is asymptotically false
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-13
    Xiaoyu He; Yuval Wigderson

    Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant δ>0 such that for all c sufficiently large, there exist graphs G and H with chromatic number at least (1+δ)c for which χ(G×H)≤c.

    更新日期:2020-04-21
  • Two Erdős–Hajnal-type theorems in hypergraphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-10
    Michal Amir; Asaf Shapira; Mykhaylo Tyomkyn

    The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs. A theorem of Rödl asserts that if an n-vertex graph is non-universal then it contains an

    更新日期:2020-04-21
  • Decompositions into isomorphic rainbow spanning trees
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-10
    Stefan Glock; Daniela Kühn; Richard Montgomery; Deryk Osthus

    A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph K2n, there exists a decomposition of K2n into isomorphic rainbow spanning trees. This settles conjectures of Brualdi–Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.

    更新日期:2020-04-21
  • The Alon-Tarsi number of a planar graph minus a matching
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-02
    Jarosław Grytczuk; Xuding Zhu

    This paper proves that every planar graph G contains a matching M such that the Alon-Tarsi number of G−M is at most 4. As a consequence, G−M is 4-paintable, and hence G itself is 1-defective 4-paintable. This improves a result of Cushing and Kierstead (2010) [5], who proved that every planar graph is 1-defective 4-choosable.

    更新日期:2020-04-21
  • Induced subgraphs of graphs with large chromatic number. VI. Banana trees
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-01-27
    Alex Scott; Paul Seymour

    We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6], the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2], we proved that every cycle has this property. Here we give a common

    更新日期:2020-04-21
  • On a perfect matching in a random digraph with average out-degree below two
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-20
    Michal Karoński, Ed Overman, Boris Pittel

    Existence of a perfect matching in a random bipartite digraph with bipartition (V1,V2), |Vi|=n, is studied. The graph is generated in two rounds of random selections of a potential matching partner such that the average number of selections made by each vertex overall is below 2. More precisely, in the first round each vertex chooses a potential mate uniformly at random, and independently of all vertices

    更新日期:2020-03-20
  • 2-factors with k cycles in Hamiltonian graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-03-10
    Matija Bucić, Erik Jahn, Alexey Pokrovskiy, Benny Sudakov

    A well known generalisation of Dirac's theorem states that if a graph G on n≥4k vertices has minimum degree at least n/2 then G contains a 2-factor consisting of exactly k cycles. This is easily seen to be tight in terms of the bound on the minimum degree. However, if one assumes in addition that G is Hamiltonian it has been conjectured that the bound on the minimum degree may be relaxed. This was

    更新日期:2020-03-10
  • Edge-critical subgraphs of Schrijver graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-02-14
    Tomáš Kaiser, Matěj Stehlík

    For k≥1 and n≥2k, the Kneser graph KG(n,k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of KG(n,k) is n−2k+2. Schrijver constructed a vertex-critical subgraph SG(n,k) of KG(n,k) with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges

    更新日期:2020-02-14
  • On the chromatic number of disjointness graphs of curves
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-02-14
    János Pach, István Tomon

    Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one

    更新日期:2020-02-14
  • 3-Flows with large support
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-02-06
    Matt DeVos, Jessica McDonald, Irene Pivotto, Edita Rollová, Robert Šámal

    We prove that every 3-edge-connected graph G has a 3-flow ϕ with the property that |supp(ϕ)|≥56|E(G)|. The graph K4 demonstrates that this 56 ratio is best possible; there is an infinite family where 56 is tight.

    更新日期:2020-02-06
  • The hat guessing number of graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-01-24
    Noga Alon, Omri Ben-Eliezer, Chong Shangguan, Itzhak Tamo

    Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they

    更新日期:2020-01-24
  • Maximum number of colourings: 4-chromatic graphs
    J. Comb. Theory B (IF 1.306) Pub Date : 2020-01-23
    Fiachra Knox, Bojan Mohar

    It is proved that every connected graph G on n vertices with χ(G)≥4 has at most k(k−1)n−3(k−2)(k−3) k-colourings for every k≥4. Equality holds for some (and then for every) k if and only if the graph is formed from K4 by repeatedly adding leaves. This confirms (a strengthening of) the 4-chromatic case of a long-standing conjecture of Tomescu [29]. Proof methods may be of independent interest. In particular

    更新日期:2020-01-23
  • Maximum degree and diversity in intersecting hypergraphs
    J. Comb. Theory B (IF 1.306) 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:2020-01-15
  • Hamiltonicity in randomly perturbed hypergraphs
    J. Comb. Theory B (IF 1.306) 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

    更新日期:2020-01-10
  • The number of Gallai k-colorings of complete graphs
    J. Comb. Theory B (IF 1.306) 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

    更新日期:2019-12-31
  • Cycles containing all the odd-degree vertices
    J. Comb. Theory B (IF 1.306) 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:2019-12-31
  • The Kelmans-Seymour conjecture IV: A proof
    J. Comb. Theory B (IF 1.306) 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

    更新日期:2019-12-19
  • Finding a path with two labels forbidden in group-labeled graphs
    J. Comb. Theory B (IF 1.306) 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

    更新日期:2019-12-16
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