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Minimal asymmetric hypergraphs J. Comb. Theory B (IF 1.4) Pub Date : 20230921
Yiting Jiang, Jaroslav NešetřilIn this paper, we prove that for any k≥3, there exist infinitely many minimal asymmetric kuniform hypergraphs. This is in a striking contrast to k=2, where it has been proved recently that there are exactly 18 minimal asymmetric graphs. We also determine, for every k≥1, the minimum size of an asymmetric kuniform hypergraph.

Disjointness graphs of short polygonal chains J. Comb. Theory B (IF 1.4) Pub Date : 20230918
János Pach, Gábor Tardos, Géza TóthThe disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χbounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G). Here we show that this statement does not remain true for systems of

Intersecting families of sets are typically trivial J. Comb. Theory B (IF 1.4) Pub Date : 20230920
József Balogh, Ramon I. Garcia, Lina Li, Adam Zsolt WagnerA family of subsets of [n] is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for n≥2k+cklnk, almost all kuniform intersecting families are stars. Improving their result, we show that the same conclusion holds for n≥2k+100lnk

Bipartite graphs with no K6 minor J. Comb. Theory B (IF 1.4) Pub Date : 20230920
Maria Chudnovsky, Alex Scott, Paul Seymour, Sophie SpirklA theorem of Mader shows that every graph with average degree at least eight has a K6 minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have K6 minors, but minimum degree six is certainly not enough. For every ε>0 there are arbitrarily

Strengthening Hadwiger's conjecture for 4 and 5chromatic graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230912
Anders Martinsson, Raphael SteinerHadwiger's famous coloring conjecture states that every tchromatic graph contains a Ktminor. Holroyd [11] conjectured the following strengthening of Hadwiger's conjecture: If G is a tchromatic graph and S⊆V(G) takes all colors in every tcoloring of G, then G contains a Ktminor rooted at S. We prove this conjecture in the first open case of t=4. Notably, our result also directly implies a stronger

Entanglements J. Comb. Theory B (IF 1.4) Pub Date : 20230913
Johannes Carmesin, Jan KurkofkaRobertson and Seymour constructed for every graph G a treedecomposition that efficiently distinguishes all the tangles in G. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit onestep construction that is canonical. The key ingredient is an axiomatisation of ‘local properties’ of tangles. Generalisations to locally finite

Excluded minors are almost fragile II: Essential elements J. Comb. Theory B (IF 1.4) Pub Date : 20230904
Nick Brettell, James Oxley, Charles Semple, Geoff WhittleLet M be an excluded minor for the class of Prepresentable matroids for some partial field P, let N be a 3connected strong Pstabilizer that is nonbinary, and suppose M has a pair of elements {a,b} such that M﹨a,b is 3connected with an Nminor. Suppose also that E(M)≥E(N)+11 and M﹨a,b is not Nfragile. In the prequel to this paper, we proved that M﹨a,b is at most five elements away from an

Determining triangulations and quadrangulations by boundary distances J. Comb. Theory B (IF 1.4) Pub Date : 20230831
John HaslegraveWe show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be determined from the pairwise graph distances between boundary vertices. A similar result holds for disc quadrangulations with all internal vertices having degree at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local nonpositive

Strengthening Rödl's theorem J. Comb. Theory B (IF 1.4) Pub Date : 20230831
Maria Chudnovsky, Alex Scott, Paul Seymour, Sophie SpirklWhat can be said about the structure of graphs that do not contain an induced copy of some graph H? Rödl showed in the 1980s that every Hfree graph has large parts that are very sparse or very dense. More precisely, let us say that a graph F on n vertices is εrestricted if either F or its complement has maximum degree at most εn. Rödl proved that for every graph H, and every ε>0, every Hfree graph

The excluded minors for 2 and 3regular matroids J. Comb. Theory B (IF 1.4) Pub Date : 20230829
Nick Brettell, James Oxley, Charles Semple, Geoff WhittleThe class of 2regular matroids is a natural generalisation of regular and nearregular matroids. We prove an excludedminor characterisation for the class of 2regular matroids. The class of 3regular matroids coincides with the class of matroids representable over the Hydra5 partial field, and the 3connected matroids in the class with a U2,5 or U3,5minor are precisely those with six inequivalent

A proof of the tree alternative conjecture under the topological minor relation J. Comb. Theory B (IF 1.4) Pub Date : 20230829
Jorge Bruno, Paul J. SzeptyckiIn 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the equivalence class of a tree under the embeddability relation is, up to isomorphism, either trivial or infinite. In 2022 Abdi, et al. provided a rigorous exposition of a counterexample to TAC developed by Tetano in his 2008 PhD thesis. In this paper we provide a positive answer to TAC for a weaker type of graph relation: the

K4intersecting families of graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230821
Aaron Berger, Yufei ZhaoEllis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sós showing that any triangleintersecting family of graphs on n vertices has size at most 2(n2)−3, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to crossintersecting families, as well to Ktintersecting families. We prove these conjectures for t∈{3,4}, showing

Codegree threshold for rainbow perfect matchings in uniform hypergraphs J. Comb. Theory B (IF 1.4) Pub Date : 20230811
Hongliang Lu, Yan Wang, Xingxing YuLet k and n be two integers, with k≥3, n≡0(modk), and n sufficiently large. We determine the (k−1)degree threshold for the existence of a rainbow perfect matchings in nvertex kuniform hypergraph. This implies the result of Rödl, Ruciński, and Szemerédi on the (k−1)degree threshold for the existence of perfect matchings in nvertex kuniform hypergraphs. In our proof, we identify the extremal configurations

Twoarctransitive bicirculants J. Comb. Theory B (IF 1.4) Pub Date : 20230726
Wei JinIn this paper, we determine the class of finite 2arctransitive bicirculants. We show that a connected 2arctransitive bicirculant is one of the following graphs: C2n where n⩾2, K2n where n⩾2, Kn,n where n⩾3, Kn,n−nK2 where n⩾4, B(PG(d−1,q)) and B′(PG(d−1,q)) where d≥3 and q is a prime power, X1(4,q) where q≡3(mod4) is a prime power, Kq+12d where q is an odd prime power and d≥2 dividing q−1, ATQ(1+q

Linear cycles of consecutive lengths J. Comb. Theory B (IF 1.4) Pub Date : 20230704
Tao Jiang, Jie Ma, Liana YepremyanA wellknown result of Verstraëte [23] shows that for each integer k≥2 every graph G with average degree at least 8k contains cycles of k consecutive even lengths, the shortest of which is of length at most twice the radius of G. We establish two extensions of Verstraëte's result for linear cycles in linear runiform hypergraphs. We show that for any fixed integers r≥3 and k≥2, there exist constants

On 2cycles of graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230622
Sergey Norin, Robin Thomas, Hein van der HolstLet G=(V,E) be a finite undirected graph. Orient the edges of G in an arbitrary way. A 2cycle on G is a function d:E2→Z such for each edge e, d(e,⋅) and d(⋅,e) are circulations on G, and d(e,f)=0 whenever e and f have a common vertex. We show that each 2cycle is a sum of three special types of 2cycles: cyclepair 2cycles, Kuratowski 2cycles, and quad 2cycles. In the case that the graph is Kuratowski

Local Hadwiger's Conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20230620
Benjamin Moore, Luke Postle, Lise TurnerWe propose local versions of Hadwiger's Conjecture, where only balls of radius Ω(log(v(G))) around each vertex are required to be Ktminorfree. We ask: if a graph is locallyKtminorfree, is it tcolourable? We show that the answer is yes when t≤5, even in the stronger setting of listcolouring, and we complement this result with a O(logv(G))round distributed colouring algorithm in the LOCAL model

Octopuses in the Boolean cube: Families with pairwise small intersections, part I J. Comb. Theory B (IF 1.4) Pub Date : 20230608
Andrey Kupavskii, Fedor NoskovLet F1,…,Fℓ be families of subsets of {1,…,n}. Suppose that for distinct k,k′ and arbitrary F1∈Fk,F2∈Fk′ we have F1∩F2⩽m. What is the maximal value of F1…Fℓ? In this work we find the asymptotic of this product as n tends to infinity for constant ℓ and m. This question is related to a conjecture of Bohn et al. that arose in the 2level polytope theory and asked for the largest product of the number

The Ramsey number of a long even cycle versus a star J. Comb. Theory B (IF 1.4) Pub Date : 20230529
Peter Allen, Tomasz Łuczak, Joanna Polcyn, Yanbo ZhangWe find the exact value of the Ramsey number R(C2ℓ,K1,n), when ℓ and n=O(ℓ10/9) are large. Our result is closely related to the behaviour of Turán number ex(N,C2ℓ) for an even cycle whose length grows quickly with N.

Onetoone correspondence between interpretations of the Tutte polynomials J. Comb. Theory B (IF 1.4) Pub Date : 20230526
Martin KocholWe study relation between two interpretations of the Tutte polynomial of a matroid perspective M1→M2 on a set E given with a linear ordering <. A well known interpretation uses internal and external activities on a family B(M1,M2) of the sets independent in M1 and spanning in M2. Recently we introduced another interpretation based on a family D(M1,M2;<) of “cyclic bases” of M1→M2 with respect to <

Orientations of goldenmean matroids J. Comb. Theory B (IF 1.4) Pub Date : 20230511
Jakayla Robbins, Daniel SlilatyBland and Las Vergnas proved that orientations of binary matroids are induced by totally unimodular representations. (A related result is due to Minty.) Lee and Scobee proved that orientations of ternary matroids are induced by dyadic representations. In this paper we prove that consistently ordered orientations of quaternary matroids are induced by goldenmean representations.

Polynomial χbinding functions for tbroomfree graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230511
Xiaonan Liu, Joshua Schroeder, Zhiyu Wang, Xingxing YuFor any positive integer t, a tbroom is a graph obtained from K1,t+1 by subdividing an edge once. In this paper, we show that, for graphs G without induced tbrooms, we have χ(G)=o(ω(G)t+1), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. When t=2, this answers a question of Schiermeyer and Randerath. Moreover, for t=2, we strengthen the bound on χ(G) to 7ω(G)2,

On Andreae's ubiquity conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20230505
Johannes CarmesinA graph H is ubiquitous if every graph G that for every natural number n contains n vertexdisjoint Hminors contains infinitely many vertexdisjoint Hminors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.

Graph product structure for nonminorclosed classes J. Comb. Theory B (IF 1.4) Pub Date : 20230502
Vida Dujmović, Pat Morin, David R. WoodDujmović et al. [J. ACM '20] proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apexminorfree graphs. These tools have been used to solve longstanding problems on queue layouts, nonrepetitive colouring, pcentered colouring, and adjacency labelling. This paper

How to build a pillar: A proof of Thomassen's conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20230426
Irene Gil Fernández, Hong LiuCarsten Thomassen in 1989 conjectured that if a graph has minimum degree much more than the number of atoms in the universe (δ(G)≥101010), then it contains a pillar, which is a graph that consists of two vertexdisjoint cycles of the same length, s say, along with s vertexdisjoint paths of the same length3 which connect matching vertices in order around the cycles. Despite the simplicity of the structure

List 4colouring of planar graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230420
Xuding ZhuThis paper proves the following result: If G is a planar graph and L is a 4list assignment of G such that L(x)∩L(y)≤2 for every edge xy, then G is Lcolourable. This answers a question asked by Kratochvíl et al. (1998) [10].

Disjoint isomorphic balanced clique subdivisions J. Comb. Theory B (IF 1.4) Pub Date : 20230407
Irene Gil Fernández, Joseph Hyde, Hong Liu, Oleg Pikhurko, Zhuo WuA classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least Ck2 has a subdivision of Kk, the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertexdisjoint isomorphic copies of a Kksubdivision

Pure pairs. VII. Homogeneous submatrices in 0/1matrices with a forbidden submatrix J. Comb. Theory B (IF 1.4) Pub Date : 20230407
Alex Scott, Paul Seymour, Sophie SpirklFor integer n>0, let f(n) be the number of rows of the largest all0 or all1 square submatrix of M, minimized over all n×n 0/1matrices M. Thus f(n)=O(logn). But let us fix a matrix H, and define fH(n) to be the same, minimized over all n×n 0/1matrices M such that neither M nor its complement (that is, change all 0's to 1's and vice versa) contains H as a submatrix. It is known that fH(n)≥εnc, where

Hypergraph Turán densities can have arbitrarily large algebraic degree J. Comb. Theory B (IF 1.4) Pub Date : 20230405
Xizhi Liu, Oleg PikhurkoGrosu (2016) [11] asked if there exist an integer r≥3 and a finite family of rgraphs whose Turán density, as a real number, has (algebraic) degree greater than r−1. In this note we show that, for all integers r≥3 and d, there exists a finite family of rgraphs whose Turán density has degree at least d, thus answering Grosu's question in a strong form.

Graphs of bounded twinwidth are quasipolynomially χbounded J. Comb. Theory B (IF 1.4) Pub Date : 20230328
Michał Pilipczuk, Marek SokołowskiWe prove that for every t∈N there is a constant γt such that every graph with twinwidth at most t and clique number ω has chromatic number bounded by 2γtlog4t+3ω. In other words, we prove that graph classes of bounded twinwidth are quasipolynomially χbounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially χbounded

Evenholefree graphs still have bisimplicial vertices J. Comb. Theory B (IF 1.4) Pub Date : 20230323
Maria Chudnovsky, Paul SeymourA hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An evenholefree graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In an earlier paper [1], AddarioBerry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed

Embedding cliquefactors in graphs with low ℓindependence number J. Comb. Theory B (IF 1.4) Pub Date : 20230323
Fan Chang, Jie Han, Jaehoon Kim, Guanghui Wang, Donglei YangThe following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: given μ>0 and integers ℓ,r and n with n∈rN, is it true that there exists an α>0 such that every nvertex graph G with δ(G)≥max{12,r−ℓr}n+μn and αℓ(G)≤αn contains a Krfactor? We give a negative answer to this question for the case ℓ≥3r4 by giving a family of constructions using the socalled cover thresholds

Coloring polygon visibility graphs and their generalizations J. Comb. Theory B (IF 1.4) Pub Date : 20230317
James Davies, Tomasz Krawczyk, Rose McCarty, Bartosz WalczakCurve pseudovisibility graphs generalize polygon and pseudopolygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudovisibility graph with clique number ω has chromatic number at most 3⋅4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudovisibility graph (considered as an ordered graph)

kapices of minorclosed graph classes. I. Bounding the obstructions J. Comb. Theory B (IF 1.4) Pub Date : 20230313
Ignasi Sau, Giannos Stamoulis, Dimitrios M. ThilikosLet G be a minorclosed graph class. We say that a graph G is a kapex of G if G contains a set S of at most k vertices such that G∖S belongs to G. We denote by Ak(G) the set of all graphs that are kapices of G. We prove that every graph in the obstruction set of Ak(G), i.e., the minorminimal set of graphs not belonging to Ak(G), has order at most 2222poly(k), where poly is a polynomial function

Packing cycles in undirected grouplabelled graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230313
Robin Thomas, Youngho YooWe prove a refinement of the flat wall theorem of Robertson and Seymour to undirected grouplabelled graphs (G,γ) where γ assigns to each edge of an undirected graph G an element of an abelian group Γ. As a consequence, we prove that Γnonzero cycles (cycles whose edge labels sum to a nonidentity element of Γ) satisfy the halfintegral ErdősPósa property, and we also recover a result of Wollan that

End spaces and treedecompositions J. Comb. Theory B (IF 1.4) Pub Date : 20230303
Marcel Koloschin, Thilo Krill, Max PitzWe present a systematic investigation into how treedecompositions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be distinguished, and characterise which subsets of ends can be displayed by a treedecomposition of finite adhesion. In particular, we show that a subset Ψ of the ends

Pure pairs. IV. Trees in bipartite graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230301
Alex Scott, Paul Seymour, Sophie SpirklIn this paper we investigate the bipartite analogue of the strong ErdősHajnal property. We prove that for every forest H and every τ with 0<τ≤1, there exists ε>0, such that if G has a bipartition (A,B) and does not contain H as an induced subgraph, and has at most (1−τ)A⋅B edges, then there is a stable set X of G with X∩A≥εA and X∩B≥εB. No graphs H except forests have this property.

An improved lower bound of P(G,L)−P(G,k) for kassignments L J. Comb. Theory B (IF 1.4) Pub Date : 20230227
Fengming Dong, Meiqiao ZhangLet G=(V,E) be a simple graph with n vertices and m edges, P(G,k) be the chromatic polynomial of G, and P(G,L) be the number of Lcolorings of G for any kassignment L. In this article, we show that when k≥m−1≥3, P(G,L)−P(G,k) is bounded below by ((k−m+1)kn−3+(k−m+3)c3kn−5)∑uv∈EL(u)∖L(v), where c≥(m−1)(m−3)8, and in particular, if G is K3free, then c≥(m−22)+2m−3. Consequently, P(G,L)≥P(G,k) whenever

Proper orientations and proper chromatic number J. Comb. Theory B (IF 1.4) Pub Date : 20230224
Yaobin Chen, Bojan Mohar, Hehui WuThe proper orientation number χ→(G) of a graph G is the minimum k such that there exists an orientation of the edges of G with all vertexoutdegrees at most k and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper orientation number are resolved. First it is shown, that χ→(G) of any planar graph G is at most 14. Secondly, it is shown that for every

Counting colorings of trianglefree graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230224
Anton Bernshteyn, Tyler Brazelton, Ruijia Cao, Akum KangBy a theorem of Johansson, every trianglefree graph G of maximum degree Δ has chromatic number at most (C+o(1))Δ/logΔ for some universal constant C>0. Using the entropy compression method, Molloy proved that one can in fact take C=1. Here we show that for every q⩾(1+o(1))Δ/logΔ, the number c(G,q) of proper qcolorings of G satisfiesc(G,q)⩾(1−1q)m((1−o(1))q)n, where n=V(G) and m=E(G). Except

A note on classes of subgraphs of locally finite graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230221
Florian LehnerWe investigate the question how ‘small’ a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph H containing all elements of a graph class G. These conditions imply that such a graph H exists for the class Gd consisting of

Characterising graphs with no subdivision of a wheel of bounded diameter J. Comb. Theory B (IF 1.4) Pub Date : 20230210
Johannes CarmesinWe prove that a graph has an rbounded subdivision of a wheel if and only if it does not have a graphdecomposition of locality r and width at most two.

Ádám's conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20230206
Carsten ThomassenWe describe an infinite family of strongly 2connected oriented graphs (that is, directed graphs with no multiple arcs) containing no arc whose reversal decreases the number of directed cycles.

Twinwidth can be exponential in treewidth J. Comb. Theory B (IF 1.4) Pub Date : 20230125
Édouard Bonnet, Hugues DéprésFor any small positive real ε and integer t>1ε, we build a graph with a vertex deletion set of size t to a tree, and twinwidth greater than 2(1−ε)t. In particular, this shows that the twinwidth is sometimes exponential in the treewidth, in the socalled oriented twinwidth and grid number, and that adding an apex may multiply the twinwidth by at least 2−ε. Except for the one in oriented twinwidth

On the central levels problem J. Comb. Theory B (IF 1.4) Pub Date : 20230120
Petr Gregor, Ondřej Mička, Torsten MützeThe central levels problem asserts that the subgraph of the (2m+1)dimensional hypercube induced by all bitstrings with at least m+1−ℓ many 1s and at most m+ℓ many 1s, i.e., the vertices in the middle 2ℓ levels, has a Hamilton cycle for any m≥1 and 1≤ℓ≤m+1. This problem was raised independently by Buck and Wiedemann, Savage, Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization

Isomorphisms between random graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230117
Sourav Chatterjee, Persi DiaconisConsider two independent Erdős–Rényi G(N,1/2) graphs. We show that with probability tending to 1 as N→∞, the largest induced isomorphic subgraph has size either ⌊xN−εN⌋ or ⌊xN+εN⌋, where xN=4log2N−2log2log2N−2log2(4/e)+1 and εN=(4log2N)−1/2. Using similar techniques, we also show that if Γ1 and Γ2 are independent G(n,1/2) and G(N,1/2) random graphs, then Γ2 contains an isomorphic copy of Γ1 as

Packing Apaths of length zero modulo a prime J. Comb. Theory B (IF 1.4) Pub Date : 20230109
Robin Thomas, Youngho YooIt is known that Apaths of length 0 mod m satisfy the ErdősPósa property if m=2 or m=4, but not if m>4 is composite. We show that if p is prime, then Apaths of length 0 mod p satisfy the ErdősPósa property. More generally, in the framework of undirected grouplabeled graphs, we characterize the abelian groups Γ and elements ℓ∈Γ for which the ErdősPósa property holds for Apaths of weight ℓ.

Tiling multipartite hypergraphs in quasirandom hypergraphs J. Comb. Theory B (IF 1.4) Pub Date : 20230103
Laihao Ding, Jie Han, Shumin Sun, Guanghui Wang, Wenling ZhouGiven k≥2 and two kgraphs (kuniform hypergraphs) F and H, an Ffactor in H is a set of vertex disjoint copies of F that together covers the vertex set of H. Lenz and Mubayi studied the Ffactor problems in quasirandom kgraphs with minimum degree Ω(nk−1). In particular, they constructed a sequence of 1/8dense quasirandom 3graphs H(n) with minimum degree Ω(n2) and minimum codegree Ω(n) but with

Turán problems for edgeordered graphs J. Comb. Theory B (IF 1.4) Pub Date : 20230103
Dániel Gerbner, Abhishek Methuku, Dániel T. Nagy, Dömötör Pálvölgyi, Gábor Tardos, Máté VizerIn this paper we initiate a systematic study of the Turán problem for edgeordered graphs. A simple graph is called edgeordered if its edges are linearly ordered. This notion allows us to study graphs (and in particular their maximum number of edges) when a subgraph is forbidden with a specific edgeorder but the same underlying graph may appear with a different edgeorder. We prove an ErdősStoneSimonovitstype

Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching J. Comb. Theory B (IF 1.4) Pub Date : 20221229
František Kardoš, Edita Máčajová, Jean Paul ZerafaLet G be a bridgeless cubic graph. The Berge–Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan–Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs

Obstructions for matroids of pathwidth at most k and graphs of linear rankwidth at most k J. Comb. Theory B (IF 1.4) Pub Date : 20221229
Mamadou Moustapha Kanté, Eun Jung Kim, Ojoung Kwon, Sangil OumEvery minorclosed class of matroids of bounded branchwidth can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field F, the list needs to contain only finitely many Frepresentable matroids, due to the wellquasiordering of Frepresentable matroids of bounded branchwidth under taking matroid minors [J


Finite 3connectedsethomogeneous locally 2Kn graphs and sarctransitive graphs J. Comb. Theory B (IF 1.4) Pub Date : 20221221
JinXin ZhouIn this paper, all graphs are assumed to be finite. For s≥1 and a graph Γ, if for every pair of isomorphic connected induced subgraphs on at most s vertices there exists an automorphism of Γ mapping the first to the second, then we say that Γ is sconnectedsethomogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at most s vertices can be extended to an automorphism

On Vizing's edge colouring question J. Comb. Theory B (IF 1.4) Pub Date : 20221219
Marthe Bonamy, Oscar Defrain, Tereza Klimošová, Aurélie Lagoutte, Jonathan NarboniSoon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for trianglefree graphs.

Primevalent symmetric graphs with a quasisemiregular automorphism J. Comb. Theory B (IF 1.4) Pub Date : 20221214
FuGang Yin, YanQuan Feng, JinXin Zhou, AHui JiaAn automorphism of a graph is called quasisemiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the wellknown problem regarding existence of semiregular automorphisms in vertextransitive graphs. Symmetric graphs of valency three

Distinct degrees and homogeneous sets J. Comb. Theory B (IF 1.4) Pub Date : 20221213
Eoin Long, Laurenţiu PloscaruIn this paper we investigate the extremal relationship between two wellstudied graph parameters: the order of the largest homogeneous set in a graph G and the maximal number of distinct degrees appearing in an induced subgraph of G, denoted respectively by hom(G) and f(G). Our main theorem improves estimates due to several earlier researchers and shows that if G is an nvertex graph with hom(G)≥n1/2

The minimum number of cliquesaturating edges J. Comb. Theory B (IF 1.4) Pub Date : 20221208
Jialin He, Fuhong Ma, Jie Ma, Xinyang YeLet G be a Kpfree graph. We say e is a Kpsaturating edge of G if e∉E(G) and G+e contains a copy of Kp. Denote by fp(n,m) the minimum number of Kpsaturating edges that an nvertex Kpfree graph with m edges can have. Erdős and Tuza conjectured that f4(n,⌊n2/4⌋+1)=(1+o(1))n216. Balogh and Liu disproved this by showing f4(n,⌊n2/4⌋+1)=(1+o(1))2n233. They believed that a natural generalization of their

The LovászCherkassky theorem in countable graphs J. Comb. Theory B (IF 1.4) Pub Date : 20221130
Attila JoóLovász and Cherkassky discovered in the 1970s independently that if G is a finite graph with a given set T of terminal vertices such that G is inner Eulerian with respect to T, then the maximal number of edgedisjoint paths connecting distinct vertices in T is ∑t∈Tλ(t,T−t) where λ is the local edgeconnectivity function. The optimality of a system of edgedisjoint Tpaths in the LovászCherkassky theorem

On a conjecture of spectral extremal problems J. Comb. Theory B (IF 1.4) Pub Date : 20221130
Jing Wang, Liying Kang, Yisai XueFor a simple graph F, let Ex(n,F) and Exsp(n,F) denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an nvertex graph without any copy of the graph F, respectively. The Turán graph Tn,r is the complete rpartite graph on n vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with

On the coequal values of total chromatic number and chromatic index J. Comb. Theory B (IF 1.4) Pub Date : 20221107
Guantao Chen, Yanli HaoThe chromatic index χ′(G) of a graph G is the least number of colors assigned to the edges of G such that no two adjacent edges receive the same color. The total chromatic number χ″(G) of a graph G is the least number of colors assigned to the edges and vertices of G such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color