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The maximum number of copies of an even cycle in a planar graph J. Comb. Theory B (IF 1.4) Pub Date : 20240222
Zequn Lv, Ervin Győri, Zhen He, Nika Salia, Casey Tompkins, Xiutao Zhu 
How connectivity affects the extremal number of trees J. Comb. Theory B (IF 1.4) Pub Date : 20240219
Suyun Jiang, Hong Liu, Nika SaliaThe ErdősSós conjecture states that the maximum number of edges in an vertex graph without a given vertex tree is at most . Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a vertex tree , we construct vertex connected graphs that are free with at least

The minimum degree removal lemma thresholds J. Comb. Theory B (IF 1.4) Pub Date : 20240126
Lior Gishboliner, Zhihan Jin, Benny SudakovThe graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε>0, if an nvertex graph G contains εn2 edgedisjoint copies of H then G contains δnv(H) copies of H for some δ=δ(ε,H)>0. The current proofs of the removal lemma give only very weak bounds on δ(ε,H), and it is also known that δ(ε,H) is not polynomial in ε unless H is bipartite. Recently

A solution to the 123 conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20240126
Ralph KeuschWe show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Łuczak, and Thomason from 2004.

The core conjecture of Hilton and Zhao J. Comb. Theory B (IF 1.4) Pub Date : 20240125
Yan Cao, Guantao Chen, Guangming Jing, Songling ShanA simple graph G with maximum degree Δ is overfull if E(G)>Δ⌊V(G)/2⌋. The core of G, denoted GΔ, is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals Δ+1 if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with Δ≥3 and Δ(GΔ)≤2, then χ′(G)=Δ+1 implies that G is overfull or G=P⁎, where P⁎ is obtained

On orders of automorphisms of vertextransitive graphs J. Comb. Theory B (IF 1.4) Pub Date : 20240123
Primož Potočnik, Micael Toledo, Gabriel VerretIn this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertextransitive graphs. In particular, we show that the order of every automorphism of a connected vertextransitive graph with n vertices and of valence d, d≤4, is at most cdn where c3=1 and c4=9. Whether such a constant cd exists for valencies larger than 4 remains an unanswered question. Further

Extended commonality of paths and cycles via Schur convexity J. Comb. Theory B (IF 1.4) Pub Date : 20240117
Jang Soo Kim, Joonkyung LeeA graph H is common if the number of monochromatic copies of H in a 2edgecolouring of the complete graph Kn is asymptotically minimised by the random colouring, or equivalently, tH(W)+tH(1−W)≥21−e(H) holds for every graphon W:[0,1]2→[0,1], where tH(.) denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to

Excluded minors for the Klein bottle II. Cascades J. Comb. Theory B (IF 1.4) Pub Date : 20240112
Bojan Mohar, Petr ŠkodaGraphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus k or for embeddings into nonorientable surface of genus k are built from 3connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete

A critical probability for biclique partition of Gn,p J. Comb. Theory B (IF 1.4) Pub Date : 20240112
Tom Bohman, Jakob HofstadThe biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G)

Count and cofactor matroids of highly connected graphs J. Comb. Theory B (IF 1.4) Pub Date : 20240105
Dániel Garamvölgyi, Tibor Jordán, Csaba KirályWe consider two types of matroids defined on the edge set of a graph G: count matroids Mk,ℓ(G), in which independence is defined by a sparsity count involving the parameters k and ℓ, and the C21cofactor matroid C(G), in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair (k,ℓ), that if G is sufficiently highly connected, then G−e has maximum rank

Sparse graphs without long induced paths J. Comb. Theory B (IF 1.4) Pub Date : 20240105
Oscar Defrain, JeanFlorent RaymondGraphs of bounded degeneracy are known to contain induced paths of order Ω(loglogn) when they contain a path of order n, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to Ω((logn)c) for some constant c>0 depending on the degeneracy. We disprove this conjecture by constructing, for arbitrarily large values of


Turán graphs with bounded matching number J. Comb. Theory B (IF 1.4) Pub Date : 20231215
Noga Alon, Péter FranklWe determine the maximum possible number of edges of a graph with n vertices, matching number at most s and clique number at most k for all admissible values of the parameters.

On a problem of ElZahar and Erdős J. Comb. Theory B (IF 1.4) Pub Date : 20231211
Tung Nguyen, Alex Scott, Paul SeymourTwo subgraphs A,B of a graph G are anticomplete if they are vertexdisjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by ElZahar and Erdős in 1986, and remains open. If so, then at least there should be two

Graph partitions under average degree constraint J. Comb. Theory B (IF 1.4) Pub Date : 20231205
Yan Wang, Hehui WuIn this paper, we prove that every graph with average degree at least s+t+2 has a vertex partition into two parts, such that one part has average degree at least s, and the other part has average degree at least t. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan.

Dimension is polynomial in height for posets with planar cover graphs J. Comb. Theory B (IF 1.4) Pub Date : 20231129
Jakub Kozik, Piotr Micek, William T. TrotterWe show that height h posets that have planar cover graphs have dimension O(h6). Previously, the best upper bound was 2O(h3). Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes K5 as a minor.

Hitting all maximum stable sets in P5free graphs J. Comb. Theory B (IF 1.4) Pub Date : 20231129
Sepehr Hajebi, Yanjia Li, Sophie SpirklWe prove that every P5free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where Pt denotes the tvertex path, and for graphs G,H, we say G is Hfree if no induced subgraph of G is isomorphic to H). More generally, let us say a class C of graphs is ηbounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest

Edgecolouring graphs with local list sizes J. Comb. Theory B (IF 1.4) Pub Date : 20231122
Marthe Bonamy, Michelle Delcourt, Richard Lang, Luke PostleThe famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree

Diractype conditions for spanning boundeddegree hypertrees J. Comb. Theory B (IF 1.4) Pub Date : 20231122
Matías PavezSigné, Nicolás SanhuezaMatamala, Maya SteinWe prove that for fixed k, every kuniform hypergraph on n vertices and of minimum codegree at least n/2+o(n) contains every spanning tight ktree of bounded vertex degree as a subgraph. This generalises a wellknown result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions

Tight bounds for divisible subdivisions J. Comb. Theory B (IF 1.4) Pub Date : 20231116
Shagnik Das, Nemanja Draganić, Raphael SteinerAlon and Krivelevich proved that for every nvertex subcubic graph H and every integer q≥2 there exists a (smallest) integer f=f(H,q) such that every Kfminor contains a subdivision of H in which the length of every subdivisionpath is divisible by q. Improving their superexponential bound, we show that f(H,q)≤212qn+8n+14q, which is optimal up to a constant multiplicative factor.

Generalized cut trees for edgeconnectivity J. Comb. Theory B (IF 1.4) Pub Date : 20231120
OnHei Solomon Lo, Jens M. SchmidtWe present three cut trees of graphs, each of them giving insights into the edgeconnectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation R on the vertex set of the graph, which generalizes GomoryHu trees. Applying these cut trees, we prove the following: • A pair of vertices {v,w} of a graph G is pendant if λ(v,w)=min{d(v)

Threecoloring trianglefree graphs on surfaces VI. 3colorability of quadrangulations J. Comb. Theory B (IF 1.4) Pub Date : 20231118
Zdeněk Dvořák, Daniel Král', Robin ThomasWe give a lineartime algorithm to decide 3colorability (and find a 3coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

Maximal matroids in weak order posets J. Comb. Theory B (IF 1.4) Pub Date : 20231117
Bill Jackson, Shinichi TanigawaLet X be a family of subsets of a finite set E. A matroid on E is called an Xmatroid if each set in X is a circuit. We develop techniques for determining when there exists a unique maximal Xmatroid in the weak order poset of all Xmatroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of

Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph J. Comb. Theory B (IF 1.4) Pub Date : 20231108
Alex Scott, Paul SeymourThe GyárfásSumner conjecture says that for every forest H and every integer k, if G is Hfree and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is Hfree if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if

Treewidth versus clique number. II. Treeindependence number J. Comb. Theory B (IF 1.4) Pub Date : 20231109
Clément Dallard, Martin Milanič, Kenny ŠtorgelIn 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw,ω)bounded. The family of (tw,ω)bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection

Induced subgraphs and tree decompositions VII. Basic obstructions in Hfree graphs J. Comb. Theory B (IF 1.4) Pub Date : 20231107
Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie SpirklWe say a class C of graphs is clean if for every positive integer t there exists a positive integer w(t) such that every graph in C with treewidth more than w(t) contains an induced subgraph isomorphic to one of the following: the complete graph Kt, the complete bipartite graph Kt,t, a subdivision of the (t×t)wall or the line graph of a subdivision of the (t×t)wall. In this paper, we adapt a method

The immersionminimal infinitely edgeconnected graph J. Comb. Theory B (IF 1.4) Pub Date : 20231108
Paul Knappe, Jan KurkofkaWe show that there is a unique immersionminimal infinitely edgeconnected graph: every such graph contains the halved Farey graph, which is itself infinitely edgeconnected, as an immersion minor. By contrast, any minimal list of infinitely edgeconnected graphs represented in all such graphs as topological minors must be uncountable.

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree J. Comb. Theory B (IF 1.4) Pub Date : 20231030
Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Sepehr Hajebi, Paweł Rzążewski, Sophie Spirkl, Kristina VuškovićThis paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k×k)wall or the line graph of

On a recolouring version of Hadwiger's conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20231030
Marthe Bonamy, Marc Heinrich, Clément LegrandDuchesne, Jonathan NarboniWe prove that for any ε>0, for any large enough t, there is a graph that admits no Ktminor but admits a (32−ε)tcolouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.

Quantum isomorphism of graphs from association schemes J. Comb. Theory B (IF 1.4) Pub Date : 20231020
Ada Chan, William J. MartinWe show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result [20] of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph

Excluded minors for the Klein bottle I. Low connectivity case J. Comb. Theory B (IF 1.4) Pub Date : 20231020
Bojan Mohar, Petr ŠkodaGraphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2vertexcut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete

Induced paths in graphs without anticomplete cycles J. Comb. Theory B (IF 1.4) Pub Date : 20231020
Tung Nguyen, Alex Scott, Paul SeymourLet us say a graph is Osfree, where s≥1 is an integer, if there do not exist s cycles of the graph that are pairwise vertexdisjoint and have no edges joining them. The structure of such graphs, even when s=2, is not well understood. For instance, until now we did not know how to test whether a graph is O2free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that O2free

Wellquasiordering Friedman ideals of finite trees proof of Robertson's magictree conjecture J. Comb. Theory B (IF 1.4) Pub Date : 20231011
Nathan Bowler, Yared NigussieApplying a recent extension (2015) of a structure theorem of Robertson, Seymour and Thomas from 1993, in this paper we establish Robertson's magictree conjecture from 1997.

On the automorphism groups of rank4 primitive coherent configurations J. Comb. Theory B (IF 1.4) Pub Date : 20231011
Bohdan KivvaThe minimal degree of a permutation group G is the minimum number of points not fixed by nonidentity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥cn for some constant c>0. In 2014, Babai proved the desired

Excluding a planar matching minor in bipartite graphs J. Comb. Theory B (IF 1.4) Pub Date : 20231004
Archontia C. Giannopoulou, Stephan Kreutzer, Sebastian WiederrechtThe notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999. In this paper we generalise basic ideas from the graph minor series by Robertson

Hamilton cycles in dense regular digraphs and oriented graphs J. Comb. Theory B (IF 1.4) Pub Date : 20231004
Allan Lo, Viresh Patel, Mehmet Akif YıldızWe prove that for every ε>0 there exists n0=n0(ε) such that every regular oriented graph on n>n0 vertices and degree at least (1/4+ε)n has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions

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