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Cycles of a given length in tournaments J. Comb. Theory B (IF 1.491) Pub Date : 20220804
Andrzej Grzesik, Daniel Král', László M. Lovász, Jan VolecWe study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let c(ℓ) be the limit of the ratio of the maximum number of cycles of length ℓ in an nvertex tournament and the expected number of cycles of length ℓ in the random nvertex tournament, when n tends to infinity. It is wellknown that c(3)=1 and c(4)=4/3. We show that c(ℓ)=1 if and only if ℓ

Reconstructing the degree sequence of a sparse graph from a partial deck J. Comb. Theory B (IF 1.491) Pub Date : 20220803
Carla Groenland, Tom Johnston, Andrey Kupavskii, Kitty Meeks, Alex Scott, Jane TanThe deck of a graph G is the multiset of cards {G−v:v∈V(G)}. Myrvold (1992) showed that the degree sequence of a graph on n≥7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can be reconstructed from any deck missing O(n/d3) cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar

Minimizing submodular functions on diamonds via generalized fractional matroid matchings J. Comb. Theory B (IF 1.491) Pub Date : 20220805
Satoru Fujishige, Tamás Király, Kazuhisa Makino, Kenjiro Takazawa, Shinichi TanigawaIn this paper we show the first polynomialtime algorithm for the problem of minimizing submodular functions on the product of diamonds of finite size. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is a

Tutte paths and long cycles in circuit graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220804
Michael C. Wigal, Xingxing YuThomassen proved that 4connected planar graphs are Hamilton connected by showing that every 2connected planar graph G contains a Tutte path P between any two given vertices, that is, every component of G−P has at most three neighbors on P. In this paper, we prove a quantitative version of this result for circuit graphs, a natural class of planar graphs which includes all 3connected planar graphs

Bounding χ by a fraction of Δ for graphs without large cliques J. Comb. Theory B (IF 1.491) Pub Date : 20220801
Marthe Bonamy, Tom Kelly, Peter Nelson, Luke PostleThe greedy coloring algorithm shows that a graph of maximum degree at most Δ has chromatic number at most Δ+1, and this is tight for cliques. Much attention has been devoted to improving this “greedy bound” for graphs without large cliques. Brooks famously proved that this bound can be improved by one if Δ≥3 and the graph contains no clique of size Δ+1. Reed's Conjecture states that the “greedy bound”

Minimal quadrangulations of surfaces J. Comb. Theory B (IF 1.491) Pub Date : 20220728
Wenzhong Liu, M.N. Ellingham, Dong YeA quadrangular embedding of a graph in a surface Σ, also known as a quadrangulation of Σ, is a cellular embedding in which every face is bounded by a 4cycle. A quadrangulation of Σ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in Σ. In this paper we determine n(Σ), the order of a minimal quadrangulation of a surface Σ, for all surfaces, both orientable and nonorientable

Counting rgraphs without forbidden configurations J. Comb. Theory B (IF 1.491) Pub Date : 20220727
József Balogh, Felix Christian Clemen, Letícia MattosOne of the major problems in combinatorics is to determine the number of runiform hypergraphs (rgraphs) on n vertices which are free of certain forbidden structures. This problem dates back to the work of Erdős, Kleitman and Rothschild, who showed that the number of Krfree graphs on n vertices is 2ex(n,Kr)+o(n2). Their work was later extended to forbidding graphs as induced subgraphs by Prömel and

Spectral extrema of Ks,tminor free graphs – On a conjecture of M. Tait J. Comb. Theory B (IF 1.491) Pub Date : 20220722
Mingqing Zhai, Huiqiu LinMinors play a key role in graph theory, and extremal problems on forbidding minors have attracted appreciable amount of interest in the past decades. In this paper, we focus on spectral extrema of Ks,tminor free graphs, and determine extremal graphs with maximum spectral radius over all Ks,tminor free graphs of sufficiently large order. This generalizes and improves several previous results. For

Ramsey number of 1subdivisions of transitive tournaments J. Comb. Theory B (IF 1.491) Pub Date : 20220622
Nemanja Draganić, David Munhá Correia, Benny Sudakov, Raphael YusterThe study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every nvertex graph with at least εn2 edges contains a 1subdivision of the complete graph on cεn vertices, resolving another old conjecture

Induced subgraphs and tree decompositions I. Evenholefree graphs of bounded degree J. Comb. Theory B (IF 1.491) Pub Date : 20220617
Tara Abrishami, Maria Chudnovsky, Kristina VuškovićTreewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph has treewidth k if it can be decomposed by a sequence of noncrossing cutsets of size at most k into pieces of size at most k+1. The study of hereditary graph classes

Upper bounds for the necklace folding problems J. Comb. Theory B (IF 1.491) Pub Date : 20220616
Endre Csóka, Zoltán L. Blázsik, Zoltán Király, Dániel LengerA necklace can be considered as a cyclic list of n red and n blue beads in an arbitrary order. In the necklace folding problem the goal is to find a large crossingfree matching of pairs of beads of different colors in such a way that there exists a “folding” of the necklace, that is a partition into two contiguous arcs, which splits the beads of any matching edge into different arcs. We give counterexamples

Sharp bounds for the chromatic number of random Kneser graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220614
Sergei Kiselev, Andrey KupavskiiGiven positive integers n⩾2k, the Kneser graph KGn,k is a graph whose vertex set is the collection of all kelement subsets of the set {1,…,n}, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of KGn,k is equal to n−2k+2. In this paper, we study the chromatic number of the random

Topological ubiquity of trees J. Comb. Theory B (IF 1.491) Pub Date : 20220610
Nathan Bowler, Christian Elbracht, Joshua Erde, J. Pascal Gollin, Karl Heuer, Max Pitz, Maximilian TeegenLet ⊲ be a relation between graphs. We say a graph G is ⊲ubiquitous if whenever Γ is a graph with nG⊲Γ for all n∈N, then one also has ℵ0G⊲Γ, where αG is the disjoint union of α many copies of G. The Ubiquity Conjecture of Andreae, a wellknown open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper

A Stallings type theorem for quasitransitive graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220608
Matthias Hamann, Florian Lehner, Babak Miraftab, Tim RühmannWe consider locally finite, connected, quasitransitive graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graphtheoretical version of Stallings' splitting theorem for multiended finitely generated groups and indeed it implies this theorem. Our result also leads to a characterisation of accessible graphs. We obtain applications

Jordanlike characterization of automorphism groups of planar graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220606
Pavel Klavík, Roman Nedela, Peter ZemanWe investigate automorphism groups of planar graphs. The main result is a complete recursive description of all abstract groups that can be realized as automorphism groups of planar graphs. The characterization is formulated in terms of inhomogeneous wreath products. In the proof, we combine techniques from combinatorics, group theory, and geometry. Our result significantly improves the Babai's description

Crossings between nonhomotopic edges J. Comb. Theory B (IF 1.491) Pub Date : 20220531
János Pach, Gábor Tardos, Géza TóthA multigraph drawn in the plane is called nonhomotopic if no pair of its edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its endvertex in the same way. Edges are allowed to intersect each other and themselves. It is easy to see that a nonhomotopic multigraph on n>1 vertices can have arbitrarily

The chromatic profile of locally bipartite graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220524
Freddie IllingworthIn 1973, Erdős and Simonovits asked whether every nvertex trianglefree graph with minimum degree greater than 1/3⋅n is 3colourable. This question initiated the study of the chromatic profile of trianglefree graphs: for each k, what minimum degree guarantees that a trianglefree graph is kcolourable. This problem has a rich history which culminated in its complete solution by Brandt and Thomassé

Trianglefree planar graphs with at most 64n0.731 3colorings J. Comb. Theory B (IF 1.491) Pub Date : 20220520
Zdeněk Dvořák, Luke PostleThomassen conjectured that trianglefree planar graphs have exponentially many 3colorings. Recently, he disproved his conjecture by providing examples of such graphs with n vertices and at most 215n/log2n 3colorings. We improve his construction, giving examples of such graphs with at most 64nlog9/23<64n0.731 3colorings. We conjecture this exponent is optimal.

The Turán number of blowups of trees J. Comb. Theory B (IF 1.491) Pub Date : 20220520
Andrzej Grzesik, Oliver Janzer, Zoltán Lóránt NagyA conjecture of Erdős from 1967 asserts that any graph on n vertices which does not contain a fixed rdegenerate bipartite graph F has at most Cn2−1/r edges, where C is a constant depending only on F. We show that this bound holds for a large family of rdegenerate bipartite graphs, including all rdegenerate blowups of trees. Our results generalise many previously proven cases of the Erdős conjecture

Concentration of maximum degree in random planar graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220520
Mihyun Kang, Michael MissethanLet P(n,m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set [n]:={1,…,n} with m=m(n) edges. We show that in the sparse regime, when m/n≤1, with high probability the maximum degree of P(n,m) takes at most two different values. In contrast, this is not true anymore in the dense regime, when m/n>1, where the maximum degree of P(n,m) is not concentrated on any subset

Edgepartitioning 3edgeconnected graphs into paths J. Comb. Theory B (IF 1.491) Pub Date : 20220518
Tereza Klimošová, Stéphan ThomasséWe show that for every ℓ, there exists dℓ such that every 3edgeconnected graph with minimum degree dℓ can be edgepartitioned into paths of length ℓ (provided that its number of edges is divisible by ℓ). This improves a result asserting that 24edgeconnectivity and high minimum degree provides such a partition. This is best possible as 3edgeconnectivity cannot be replaced by 2edge connectivity

On the hat guessing number of a planar graph class J. Comb. Theory B (IF 1.491) Pub Date : 20220516
Peter BradshawThe hat guessing number is a graph invariant based on a hat guessing game introduced by Winkler. Using a new vertex decomposition argument involving an edge density theorem of Erdős for hypergraphs, we show that the hat guessing number of all outerplanar graphs is less than 2125000. We also define the class of layered planar graphs, which contains outerplanar graphs, and we show that every layered

Structure in sparse kcritical graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220516
Ronald J. Gould, Victor Larsen, Luke PostleRecently, Kostochka and Yancey [7] proved that a conjecture of Ore is asymptotically true by showing that every kcritical graph satisfies E(G)≥⌈(k2−1k−1)V(G)−k(k−3)2(k−1)⌉. They also characterized [8] the class of graphs that attain this bound and showed that it is equivalent to the set of kOre graphs. We show that for any k≥33 there exists an ε>0 so that if G is a kcritical graph, then E(

Spanning trees in dense directed graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220516
Amarja Kathapurkar, Richard MontgomeryIn 2001, Komlós, Sárközy and Szemerédi proved that, for each α>0, there is some c>0 and n0 such that, if n≥n0, then every nvertex graph with minimum degree at least (1/2+α)n contains a copy of every nvertex tree with maximum degree at most cn/logn. We prove the corresponding result for directed graphs. That is, for each α>0, there is some c>0 and n0 such that, if n≥n0, then every nvertex directed

Local 2separators J. Comb. Theory B (IF 1.491) Pub Date : 20220510
Johannes CarmesinHow can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing ‘Local Separators’ of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2separators analogous to the 2separator theorem; 2. an exact characterisation of graphs

Overfullness of critical class 2 graphs with a small core degree J. Comb. Theory B (IF 1.491) Pub Date : 20220510
Yan Cao, Guantao Chen, Songling ShanLet G be a simple graph, and let n, Δ(G) and χ′(G) be the order, the maximum degree and the chromatic index of G, respectively. We call G overfull if E(G)/⌊n/2⌋>Δ(G), and critical if χ′(H)<χ′(G) for every proper subgraph H of G. Clearly, if G is overfull then χ′(G)=Δ(G)+1 by Vizing's Theorem. The core of G, denoted by GΔ, is the subgraph of G induced by all its maximum degree vertices. Hilton and

Countably determined ends and graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220503
Jan Kurkofka, Ruben MelcherThe directions of an infinite graph G are a tanglelike description of its ends: they are choice functions that choose a component of G−X for all finite vertex sets X⊆V(G) in a compatible manner. Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably

Almost all optimally coloured complete graphs contain a rainbow Hamilton path J. Comb. Theory B (IF 1.491) Pub Date : 20220503
Stephen Gould, Tom Kelly, Daniela Kühn, Deryk OsthusA subgraph H of an edgecoloured graph is called rainbow if all of the edges of H have different colours. In 1989, Andersen conjectured that every proper edgecolouring of Kn admits a rainbow path of length n−2. We show that almost all optimal edgecolourings of Kn admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture

Binary scalar products J. Comb. Theory B (IF 1.491) Pub Date : 20220420
Andrey Kupavskii, Stefan WeltgeLet A,B⊆Rd both span Rd such that 〈a,b〉∈{0,1} holds for all a∈A, b∈B. We show that A⋅B≤(d+1)2d. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2level polytopes. Such polytopes have the property that for every facetdefining hyperplane H there is a parallel hyperplane H′ such that H∪H′ contain all vertices. The authors conjectured

A simple proof of the Map Color Theorem for nonorientable surfaces J. Comb. Theory B (IF 1.491) Pub Date : 20220406
Vladimir P. KorzhikWe give a more simple proof of the Map Color Theorem for nonorientable surfaces that uses only four constructions of current graphs instead of 12 constructions used in the previous proof. For every i=0,1,2,3, using an index one current graph with cyclic current group, we construct a nonorientable triangular embedding of K12s+3i+1 that can be easily modified into a nonorientable triangular embedding

Optimal oriented diameter of graphs with diameter 3 J. Comb. Theory B (IF 1.491) Pub Date : 20220405
Xiaolin Wang, Yaojun ChenLet f(d) be the smallest value for which every bridgeless graph G with diameter d admits a strong orientation G⇀ such that the diameter of G⇀ is at most f(d). Chvátal and Thomassen (JCTB, 1978) established general bounds for f(d) and proved that f(2)=6. Kwok et al. (JCTB, 2010) showed that 9≤f(3)≤11. In this paper, we determine that f(3)=9.

Embedding connected factorizations J. Comb. Theory B (IF 1.491) Pub Date : 20220404
Amin Bahmanian, Anna Johnsen, Stefan NapirataFor r:=(r1,…,rq), an rfactorization of the complete λfold huniform mvertex hypergraph λKmh is a partition of the edges of λKmh into F1,…,Fq such that each color class Fi is riregular and spanning. We prove two results on embedding factorizations. Previously, these results were only known for a few small values of h, and even then only partially. We show that for n⩾hm and s:=(s1,…,sk), the obvious

Typical and extremal aspects of friendsandstrangers graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220315
Noga Alon, Colin Defant, Noah KravitzGiven graphs X and Y with vertex sets V(X) and V(Y) of the same cardinality, the friendsandstrangers graph FS(X,Y) is the graph whose vertex set consists of all bijections σ:V(X)→V(Y), where two bijections σ and σ′ are adjacent if they agree everywhere except for two adjacent vertices a,b∈V(X) such that σ(a) and σ(b) are adjacent in Y. The most fundamental question that one can ask about these f

Diractype theorems in random hypergraphs J. Comb. Theory B (IF 1.491) Pub Date : 20220315
Asaf Ferber, Matthew KwanFor positive integers d0 and any “not too small” p, we prove that a random kuniform hypergraph G with n vertices and edge probability p typically has the property that every spanning subgraph of G with minimum ddegree at least (1+ε)md(k,n)p has a perfect matching. One interesting aspect of our proof is a “nonconstructive” application of the absorbing method, which allows us to prove a bound in terms

Colouring graphs with sparse neighbourhoods: Bounds and applications J. Comb. Theory B (IF 1.491) Pub Date : 20220310
Marthe Bonamy, Thomas Perrett, Luke PostleLet G be a graph with chromatic number χ, maximum degree Δ and clique number ω. Reed's conjecture states that χ≤⌈(1−ε)(Δ+1)+εω⌉ for all ε≤1/2. It was shown by King and Reed that, provided Δ is large enough, the conjecture holds for ε≤1/130,000. In this article, we show that the same statement holds for ε≤1/26, thus making a significant step towards Reed's conjecture. We derive this result from a general

Counting Hamiltonian cycles in planar triangulations J. Comb. Theory B (IF 1.491) Pub Date : 20220307
Xiaonan Liu, Zhiyu Wang, Xingxing YuHakimi, Schmeichel, and Thomassen (1979) [10] conjectured that every 4connected planar triangulation G on n vertices has at least 2(n−2)(n−4) Hamiltonian cycles, with equality if and only if G is a double wheel. In this paper, we show that every 4connected planar triangulation on n vertices has Ω(n2) Hamiltonian cycles. Moreover, we show that if G is a 4connected planar triangulation on n vertices

Tilings in vertex ordered graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220303
József Balogh, Lina Li, Andrew TreglownOver recent years there has been much interest in both Turán and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect Htiling in an ordered graph. In the (unordered) graph

Matrix representations of frame and liftedgraphic matroids correspond to gain functions J. Comb. Theory B (IF 1.491) Pub Date : 20220303
Daryl Funk, Irene Pivotto, Daniel SlilatyLet M be a 3connected matroid and let F be a field. Let A be a matrix over F representing M and let (G,B) be a biased graph representing M. We characterize the relationship between A and (G,B), settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation (G,B) of M, A is projectively equivalent to a canonical matrix representation arising

Trees with few leaves in tournaments J. Comb. Theory B (IF 1.491) Pub Date : 20220302
Alistair Benford, Richard MontgomeryWe prove that there exists C>0 such that any (n+Ck)vertex tournament contains a copy of every nvertex oriented tree with k leaves, improving the previously best known bound of n+O(k2) vertices to give a result tight up to the value of C. Furthermore, we show that, for each k, there exists n0, such that, whenever n⩾n0, any (n+k−2)vertex tournament contains a copy of every nvertex oriented tree with

Globally rigid powers of graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220225
Tibor Jordán, Shinichi TanigawaThe characterization of rigid graphs in Rd for d≥3 is a major open problem in rigidity theory. The same holds for globally rigid graphs. In this paper our goal is to give necessary and/or sufficient conditions for the (global) rigidity of the square G2 (and more generally, the power Gk) of a graph G in Rd, for some values of k,d. Our work is motivated by some results and conjectures of M. Cheung and

Extremal problems for pairs of triangles J. Comb. Theory B (IF 1.491) Pub Date : 20220224
Zoltán Füredi, Dhruv Mubayi, Jason O'Neill, Jacques VerstraëteA convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two distinct triangles. These were studied at length by Braß [6] (2004) and by Aronov, Dujmović, Morin, Ooms, and da Silveira [2] (2019). We determine the extremal functions exactly for seven of the eight configurations. The

An extremal problem motivated by trianglefree strongly regular graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220221
Alexander RazborovWe introduce the following combinatorial problem. Let G be a trianglefree regular graph with edge density ρ. (In this paper all densities are normalized by n,n22 etc. rather than by n−1,(n2),…) What is the minimum value a(ρ) for which there always exist two nonadjacent vertices such that the density of their common neighbourhood is ≤a(ρ)? We prove a variety of upper bounds on the function a(ρ) that

Asymptotic equivalence of Hadwiger's conjecture and its odd minorvariant J. Comb. Theory B (IF 1.491) Pub Date : 20220210
Raphael SteinerHadwiger's conjecture states that every Ktminor free graph is (t−1)colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd Ktminor is (t−1)colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form

Cyclic connectivity, edgeelimination, and the twisted Isaacs graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220203
Roman Nedela, Martin ŠkovieraEdgeelimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from

A polynomial version of Cereceda's conjecture J. Comb. Theory B (IF 1.491) Pub Date : 20220202
Nicolas Bousquet, Marc HeinrichLet k and d be positive integers such that k≥d+2. Consider two kcolourings of a ddegenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper colouring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist

Discrepancies of spanning trees and Hamilton cycles J. Comb. Theory B (IF 1.491) Pub Date : 20220128
Lior Gishboliner, Michael Krivelevich, Peleg MichaeliWe study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the rcolour spanningtree discrepancy of a graph G is equal, up to a constant, to the minimum s such that G can be separated into r equal parts by deleting s vertices. This result arguably resolves the question of estimating the spanningtree discrepancy

A global decomposition theorem for excluding immersions in graphs with no edgecut of order three J. Comb. Theory B (IF 1.491) Pub Date : 20220128
ChunHung LiuA graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if G contains H as an immersion, then for every integer k, the number of vertices of degree at least k in G is at least the number of vertices of degree at least k in H. In this

Characterization of 4critical trianglefree toroidal graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220129
Zdeněk Dvořák, Jakub PekárekWe give an exact characterization of 3colorability of trianglefree graphs drawn in the torus, in the form of 186 “templates” (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3colorable if and only if it contains a subgraph matching one of the templates. As a consequence, we show every trianglefree graph drawn in the torus with edgewidth

On sensitivity in bipartite Cayley graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220120
Ignacio GarcíaMarco, Kolja KnauerHuang proved that every set of more than half the vertices of the ddimensional hypercube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs

Hamiltonian cycles and 1factors in 5regular graphs J. Comb. Theory B (IF 1.491) Pub Date : 20220120
Nico Van Cleemput, Carol T. ZamfirescuIt is proven that for any integer g≥0 and k∈{0,…,10}, there exist infinitely many 5regular graphs of genus g containing a 1factorisation with exactly k pairs of 1factors that are perfect, i.e. form a hamiltonian cycle. For g=0 and k=10, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's “marriage” operation, we discuss two gluing techniques aimed at producing graphs of

On the abstract chromatic number and its computability for finitely axiomatizable theories J. Comb. Theory B (IF 1.491) Pub Date : 20220114
Leonardo N. CoreglianoThe celebrated Erdős–Stone–Simonovits theorem characterizes the asymptotic maximum edge density in Ffree graphs as 1−1/(χ(F)−1)+o(1), where χ(F) is the minimum chromatic number of a graph in F. In [6, Examples 25 and 31], it was shown that this result can be extended to the general setting of graphs with extra structure: the asymptotic maximum edge density of a graph with extra structure without some

Linking four vertices in graphs of large connectivity J. Comb. Theory B (IF 1.491) Pub Date : 20220107
Koyo HayashiOne of the most fundamental results in structural graph theory is the “twopaths theorem” that characterizes 2linkage by planarity. As an extension of the theorem, we consider the following problem for a fixed graph H with four vertices: Given a graph G and an injective map from V(H) to V(G), is there a subdivision of H in G with four branch vertices specified by the map? Hence the case H=2K2 corresponds

Jumps in speeds of hereditary properties in finite relational languages J. Comb. Theory B (IF 1.491) Pub Date : 20220104
M.C. Laskowski, C.A. TerryGiven a finite relational language L, a hereditary Lproperty is a class H of finite Lstructures closed under isomorphism and substructure. The speed of H is the function which sends an integer n≥1 to the number of distinct elements in H with underlying set {1,...,n}. In this paper we give a description of many new jumps in the possible speeds of a hereditary Lproperty, where L is any finite relational

Canonical double covers of circulants J. Comb. Theory B (IF 1.491) Pub Date : 20211228
Blas Fernandez, Ademir HujdurovićThe canonical double cover B(X) of a graph X is the direct product of X and K2. If Aut(B(X))≅Aut(X)×Z2 then X is called stable; otherwise X is called unstable. An unstable graph is nontrivially unstable if it is connected, nonbipartite and distinct vertices have different neighborhoods. A circulant is a Cayley graph on a cyclic group. Qin et al. (2019) [18] conjectured that there are no nontrivially

Clean tangled clutters, simplices, and projective geometries J. Comb. Theory B (IF 1.491) Pub Date : 20211228
Ahmad Abdi, Gérard Cornuéjols, Matt SuperdockA clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture. Let C be a clean tangled clutter. It was recently proved that C has a fractional packing

Extremal problems for multigraphs J. Comb. Theory B (IF 1.491) Pub Date : 20211223
A. Nicholas Day, Victor FalgasRavry, Andrew TreglownAn (n,s,q)graph is an nvertex multigraph in which every sset of vertices spans at most q edges. Turántype questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in (n,s,q)graphs. We give a general lower

Constructions of new qcryptomorphisms J. Comb. Theory B (IF 1.491) Pub Date : 20211217
Eimear Byrne, Michela Ceria, Relinde JurriusIn the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A qmatroid is a qanalogue of a matroid where subspaces play the role of the subsets in the classical theory. In this article we establish cryptomorphisms of qmatroids. In doing so we highlight the difference between classical theory and

Hamiltonian cycles above expectation in rgraphs and quasirandom rgraphs J. Comb. Theory B (IF 1.491) Pub Date : 20211217
Raphael YusterLet Hr(n,p) denote the maximum number of Hamiltonian cycles in an nvertex rgraph with density p∈(0,1). The expected number of Hamiltonian cycles in the random rgraph model Gr(n,p) is E(n,p)=pn(n−1)!/2 and in the random graph model Gr(n,m) with m=p(nr) it is, in fact, slightly smaller than E(n,p). For graphs, H2(n,p) is proved to be only larger than E(n,p) by a polynomial factor and it is an open

Minimum degree thresholds for Hamilton (k/2)cycles in kuniform hypergraphs J. Comb. Theory B (IF 1.491) Pub Date : 20211208
Hiệp Hàn, Jie Han, Yi ZhaoFor any even integer k≥6, integer d such that k/2≤d≤k−1, and sufficiently large n∈(k/2)N, we find a tight minimum ddegree condition that guarantees the existence of a Hamilton (k/2)cycle in every kuniform hypergraph on n vertices. When n∈kN, the degree condition coincides with the one for the existence of perfect matchings provided by Rödl, Ruciński and Szemerédi (for d=k−1) and Treglown and Zhao

Density of C−4critical signed graphs J. Comb. Theory B (IF 1.491) Pub Date : 20211124
Reza Naserasr, Lan Anh Pham, Zhouningxin WangA signed bipartite (simple) graph (G,σ) is said to be C−4critical if it admits no homomorphism to C−4 (a negative 4cycle) but each of its proper subgraphs does. To motivate the study of C−4critical signed graphs, we show that the notion of 4coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to C−4. In particular, the 4color theorem