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Density of C−4critical signed graphs J. Comb. Theory B (IF 1.317) 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

Nowherezero 3flows in toroidal graphs J. Comb. Theory B (IF 1.317) Pub Date : 20211123
Jiaao Li, Yulai Ma, Zhengke Miao, Yongtang Shi, Weifan Wang, CunQuan ZhangTutte's 3flow conjecture states that every 4edgeconnected graph admits a nowherezero 3flow. The planar case of Tutte's 3flow conjecture is the classical Grötzsch's Theorem (1959). Steinberg and Younger (1989) further verified Tutte's 3flow conjecture for projective planar graphs. In this paper we confirm Tutte's 3flow conjecture for all toroidal graphs.

Gray codes and symmetric chains J. Comb. Theory B (IF 1.317) Pub Date : 20211115
Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, Kaja WilleWe consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1−ℓ,n+ℓ], where 1≤ℓ≤n+1, by flipping a single bit in each step. This is a farranging generalization of the wellknown middle two levels problem (the case ℓ=1). We provide a solution for the case ℓ=2, and we solve a relaxed version of the problem for general values of ℓ

Dichromatic number and forced subdivisions J. Comb. Theory B (IF 1.317) Pub Date : 20211112
Lior Gishboliner, Raphael Steiner, Tibor SzabóWe investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph F, denote by maderχ→(F) the smallest integer k such that every kdichromatic digraph contains a subdivision of F. As our first main result, we prove that if F is an orientation of a cycle then maderχ→(F)=v(F). This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura

Extremal problems of Erdős, Faudree, Schelp and Simonovits on paths and cycles J. Comb. Theory B (IF 1.317) Pub Date : 20211029
Binlong Li, Jie Ma, Bo NingFor positive integers n>d≥k, let ϕ(n,d,k) denote the least integer ϕ such that every nvertex graph with at least ϕ vertices of degree at least d contains a path on k+1 vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function ϕ(n,d,k), and conjectured that for any positive integers n>d≥k, it holds that ϕ(n,d,k)≤⌊k−12⌋⌊nd+1⌋+ϵ, where ϵ=1 if k is odd and ϵ=2

A fast distributed algorithm for (Δ + 1)edgecoloring J. Comb. Theory B (IF 1.317) Pub Date : 20211028
Anton BernshteynWe present a deterministic distributed algorithm in the LOCAL model that finds a proper (Δ+1)edgecoloring of an nvertex graph of maximum degree Δ in poly(Δ,logn) rounds. This is the first nontrivial distributed edgecoloring algorithm that uses only Δ+1 colors (matching the bound given by Vizing's theorem). Our approach is inspired by the recent proof of the measurable version of Vizing's theorem

Degeneracy of Ptfree and C⩾tfree graphs with no large complete bipartite subgraphs J. Comb. Theory B (IF 1.317) Pub Date : 20211028
Marthe Bonamy, Nicolas Bousquet, Michał Pilipczuk, Paweł Rzążewski, Stéphan Thomassé, Bartosz WalczakA hereditary class of graphs G is χbounded if there exists a function f such that every graph G∈G satisfies χ(G)⩽f(ω(G)), where χ(G) and ω(G) are the chromatic number and the clique number of G, respectively. As one of the first results about χbounded classes, Gyárfás proved in 1985 that if G is Ptfree, i.e., does not contain a tvertex path as an induced subgraph, then χ(G)⩽(t−1)ω(G)−1. In 2017

Extremal graphs for edge blowup of graphs J. Comb. Theory B (IF 1.317) Pub Date : 20211028
LongTu YuanGiven a graph H and an integer p, the edge blowup Hp+1 of H is the graph obtained from replacing each edge in H by a clique of order p+1 where the new vertices of the cliques are all distinct. The Turán numbers for edge blowup of matchings were first studied by Erdős and Moon. In this paper, we determine the range of the Turán numbers for edge blowup of all bipartite graphs and the exact Turán numbers

The generalised Oberwolfach problem J. Comb. Theory B (IF 1.317) Pub Date : 20211021
Peter Keevash, Katherine StadenWe prove that any quasirandom dense large graph in which all degrees are equal and even can be decomposed into any given collection of twofactors (2regular spanning subgraphs). A special case of this result gives a new solution to the Oberwolfach problem.

Graphs with polynomially many minimal separators J. Comb. Theory B (IF 1.317) Pub Date : 20211021
Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Stéphan Thomassé, Nicolas Trotignon, Kristina VuškovićWe show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent

The threshold bias of the cliquefactor game J. Comb. Theory B (IF 1.317) Pub Date : 20211014
Anita Liebenau, Rajko NenadovLet r≥4 be an integer and consider the following game on the complete graph Kn for n∈rZ: Two players, Maker and Breaker, alternately claim previously unclaimed edges of Kn such that in each turn Maker claims one and Breaker claims b∈N edges. Maker wins if her graph contains a Krfactor, that is a collection of n/r vertexdisjoint copies of Kr, and Breaker wins otherwise. In other words, we consider

Triangle resilience of the square of a Hamilton cycle in random graphs J. Comb. Theory B (IF 1.317) Pub Date : 20211008
Manuela Fischer, Nemanja Škorić, Angelika Steger, Miloš TrujićSince first introduced by Sudakov and Vu in 2008, the study of resilience problems in random graphs received a lot of attention in probabilistic combinatorics. Of particular interest are resilience problems of spanning structures. It is known that for spanning structures which contain many triangles, local resilience cannot prevent an adversary from destroying all copies of the structure by removing

A lower bound on the average size of a connected vertex set of a graph J. Comb. Theory B (IF 1.317) Pub Date : 20211007
Andrew VinceThe topic is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983, Jamison proved that the average order of a subtree, over all trees of order n, is minimized by the path Pn. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the average order of a connected induced subgraph over all

The binary matroids with no odd circuits of size exceeding five J. Comb. Theory B (IF 1.317) Pub Date : 20211005
Carolyn Chun, James Oxley, Kristen WetzlerGeneralizing a graphtheoretical result of Maffray to binary matroids, Oxley and Wetzler proved that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is isomorphic to M(K4) or F7, or M is the cycle matroid of a graph consisting of a collection of triangles all sharing a common edge. In this paper, we show that if M is a 3connected binary matroid

Extremal graphs for the Tutte polynomial J. Comb. Theory B (IF 1.317) Pub Date : 20211005
Nathan KahlA graph transformation called the compression of a graph G is known to decrease the number of spanning trees, the allterminal reliability, and the magnitude of the coefficients of the chromatic polynomial of a graph G. All of these graph parameters can be derived from the Tutte polynomial of G, and in this paper we determine more generally compression's effect on the Tutte polynomial, recovering the

On the Cayleyness of PraegerXu graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210929
R. Jajcay, P. Potočnik, S. WilsonThis paper discusses a family of graphs, called PraegerXu graphs and denoted PX(n,k) here, introduced by C.E. Praeger and M.Y. Xu in 1989. These tetravalent graphs are distinguished by having large symmetry groups; their vertexstabilizers can be arbitrarily larger than the number of vertices in the graph. This paper does the following: (1) exhibits a connection between vertextransitive groups of

Cliquewidth of point configurations J. Comb. Theory B (IF 1.317) Pub Date : 20210921
Onur Çağırıcı, Petr Hliněný, Filip Pokrývka, Abhisekh SankaranWhile structural width parameters (of the input) belong to the standard toolbox of graph algorithms, it is not the usual case in computational geometry. As a case study we propose a natural extension of the structural graph parameter of cliquewidth to geometric point configurations represented by their order type. We study basic properties of this cliquewidth notion, and show that it is aligned with

Clustered variants of Hajós' conjecture J. Comb. Theory B (IF 1.317) Pub Date : 20210921
ChunHung Liu, David R. WoodHajós conjectured that every graph containing no subdivision of the complete graph Ks+1 is properly scolorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is Ω(s2/logs). We prove that O(s) colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (socalled clustered coloring). Our approach

Canonical trees of treedecompositions J. Comb. Theory B (IF 1.317) Pub Date : 20210914
Johannes Carmesin, Matthias Hamann, Babak MiraftabWe prove that every graph has a canonical tree of treedecompositions that distinguishes all principal tangles (these include the ends and various kinds of large finite dense structures) efficiently. Here ‘trees of treedecompositions’ are a slightly weaker notion than ‘treedecompositions’ but much more wellbehaved than ‘treelike metric spaces’. This theorem is best possible in the sense that we

Isomorphic bisections of cubic graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210831
S. Das, A. Pokrovskiy, B. SudakovGraph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring

Ndetachable pairs in 3connected matroids III: The theorem J. Comb. Theory B (IF 1.317) Pub Date : 20210827
Nick Brettell, Geoff Whittle, Alan WilliamsLet M be a 3connected matroid, and let N be a 3connected minor of M. A pair {x1,x2}⊆E(M) is Ndetachable if one of the matroids M/x1/x2 or M﹨x1﹨x2 is 3connected and has an Nminor. This is the third and final paper in a series where we prove that if E(M)−E(N)≥10, then either M has an Ndetachable pair after possibly performing a single ΔY or YΔ exchange, or M is essentially N with a spike

On the spectrum of Hamiltonian cycles in the ncube J. Comb. Theory B (IF 1.317) Pub Date : 20210827
A.L. PerezhoginThe spectrum of a Hamiltonian cycle (Gray code) in a Boolean ncube is a sequence of n numbers, where the ith number is equal to the number of edges of the ith direction in the cycle. Necessary conditions for the existence of a Gray code with a given spectrum are known: all numbers are even and the sum of any k numbers is at least 2k, k=1,…,n. It is proved that for all dimensions n these necessary

ErdősHajnal for capfree graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210806
Maria Chudnovsky, Paul SeymourA “cap” in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the “house” is the smallest, on five vertices. It is not known whether there exists ε>0 such that every graph G containing no house has a clique or stable set of cardinality at least Gε; this is

A proof of the upper matching conjecture for large graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210805
Ewan Davies, Matthew Jenssen, Will PerkinsWe prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every d and every large enough n divisible by 2d, a union of n/(2d) copies of the complete dregular bipartite graph maximizes the number of independent sets and matchings

The inverse KazhdanLusztig polynomial of a matroid J. Comb. Theory B (IF 1.317) Pub Date : 20210730
Alice L.L. Gao, Matthew H.Y. XieIn analogy with the classical KazhdanLusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of KazhdanLusztig polynomials for matroids. It is known that both the classical KazhdanLusztig polynomials and the matroid KazhdanLusztig polynomials can be considered as special cases of the KazhdanLusztigStanley polynomials for locally finite posets. In the framework

A characterization of Johnson and Hamming graphs and proof of Babai's conjecture J. Comb. Theory B (IF 1.317) Pub Date : 20210726
Bohdan KivvaOne of the central results in the representation theory of distanceregular graphs classifies distanceregular graphs with μ≥2 and second largest eigenvalue θ1=b1−1. In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ1≥(1−ε)b1 for the class of geometric distanceregular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism

The evolution of the structure of ABCminimal trees J. Comb. Theory B (IF 1.317) Pub Date : 20210724
Seyyed Aliasghar Hosseini, Bojan Mohar, Mohammad Bagher AhmadiThe atombond connectivity (ABC) index is a degreebased molecular descriptor that found diverse chemical applications. Characterizing trees with minimum ABCindex remained an elusive open problem even after serious attempts and is considered by some as one of the most intriguing open problems in mathematical chemistry. In this paper, we describe the exact structure of the extremal trees with sufficiently

Threecoloring trianglefree graphs on surfaces VII. A lineartime algorithm J. Comb. Theory B (IF 1.317) Pub Date : 20210724
Zdeněk Dvořák, Daniel Král', Robin ThomasWe give a lineartime algorithm to decide 3colorability of a trianglefree graph embedded in a fixed surface, and a quadratictime algorithm to output a 3coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.

Unavoidable hypergraphs J. Comb. Theory B (IF 1.317) Pub Date : 20210714
Matija Bucić, Nemanja Draganić, Benny Sudakov, Tuan TranThe following very natural problem was raised by Chung and Erdős in the early 80's and has since been repeated a number of times. What is the minimum of the Turán number ex(n,H) among all rgraphs H with a fixed number of edges? Their actual focus was on an equivalent and perhaps even more natural question which asks what is the largest size of an rgraph that can not be avoided in any rgraph on n

Enlarging vertexflames in countable digraphs J. Comb. Theory B (IF 1.317) Pub Date : 20210713
Joshua Erde, J. Pascal Gollin, Attila JoóA rooted digraph is a vertexflame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertexflame and large, where the latter means that it preserves the local connectivity to each vertex from the root

On edgeprimitive 3arctransitive graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210713
Michael Giudici, Carlisle S.H. KingThis paper begins the classification of all edgeprimitive 3arctransitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertexstabiliser acting faithfully on the set of neighbours.

On the cop number of toroidal graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210707
Florian LehnerWe show that the cop number of toroidal graphs is at most 3. This resolves a conjecture by Schroeder from 2001 which is implicit in a question by Andreae from 1986.

Edgecritical subgraphs of Schrijver graphs II: The general case J. Comb. Theory B (IF 1.317) Pub Date : 20210703
Tomáš Kaiser, Matěj StehlíkWe give a simple combinatorial description of an (n−2k+2)chromatic edgecritical subgraph of the Schrijver graph SG(n,k), itself an induced vertexcritical subgraph of the Kneser graph KG(n,k). This extends the main result of Kaiser and Stehlík (2020) [5] to all values of k, and sharpens the classical results of Lovász and Schrijver from the 1970s.

On the lower bound of the sum of the algebraic connectivity of a graph and its complement J. Comb. Theory B (IF 1.317) Pub Date : 20210630
Mostafa Einollahzadeh, Mohammad Mahdi KarkhaneeiFor a graph G, let μ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that μ2(G)+μ2(G‾)≥1, where G‾ is the complement of G. This conjecture has been proved for various families of graphs. Here, we prove this conjecture in the general case. Also, we will show that max{μ2(G),μ2(G‾)}≥1−O(n−13), where n is the number of vertices of G.

The Farey graph is uniquely determined by its connectivity J. Comb. Theory B (IF 1.317) Pub Date : 20210629
Jan KurkofkaWe show that, up to minorequivalence, the Farey graph is the unique minorminimal graph that is infinitely edgeconnected but such that every two vertices can be finitely separated.

Packing and covering immersions in 4edgeconnected graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210625
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. In this paper, we prove an edgevariant of the ErdősPósa property with respect to the immersion containment in 4edgeconnected graphs. More precisely, we prove that for every graph H, there exists a function f such that for every 4edgeconnected graph

Polynomial bounds for centered colorings on proper minorclosed graph classes J. Comb. Theory B (IF 1.317) Pub Date : 20210623
Michał Pilipczuk, Sebastian SiebertzFor p∈N, a coloring λ of the vertices of a graph G is pcentered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. Centered colorings play an important role in the theory of sparse graph classes introduced by Nešetřil and Ossona de Mendez [31], [32], as they structurally characterize classes of bounded expansion

Corrigendum to “A local epsilon version of Reed's Conjecture” [J. Combin. Theory Ser. B 141 (2020) 181–222] J. Comb. Theory B (IF 1.317) Pub Date : 20210623
Tom Kelly, Luke PostleWe correct an error that appears in Kelly and Postle (2020) [2]. All of the main results remain valid after this correction.

On the unavoidability of oriented trees J. Comb. Theory B (IF 1.317) Pub Date : 20210621
François Dross, Frédéric HavetA digraph is nunavoidable if it is contained in every tournament of order n. We first prove that every arborescence of order n with k leaves is (n+k−1)unavoidable. We then prove that every oriented tree of order n (n≥2) with k leaves is (32n+32k−2)unavoidable and (92n−52k−92)unavoidable, and thus (218n−4716)unavoidable. Finally, we prove that every oriented tree of order n with k leaves is (n

From χ to χpbounded classes J. Comb. Theory B (IF 1.317) Pub Date : 20210616
Yiting Jiang, Jaroslav Nešetřil, Patrice Ossona de Mendezχbounded classes are studied here in the context of star colorings and, more generally, χpcolorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χpboundedness

On the intersection conjecture for infinite trees of matroids J. Comb. Theory B (IF 1.317) Pub Date : 20210615
Nathan Bowler, Johannes CarmesinUsing a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2separations into finite parts.

Counterexamples to a conjecture of Erdős, Pach, Pollack and Tuza J. Comb. Theory B (IF 1.317) Pub Date : 20210611
Éva Czabarka, Inne Singgih, László A. SzékelyErdős et al. (1989) [4] conjectured that the diameter of a K2rfree connected graph of order n and minimum degree δ≥2 is at most 2(r−1)(3r+2)(2r2−1)⋅nδ+O(1) for every r≥2, if δ is a multiple of (r−1)(3r+2). For every r>1 and δ≥2(r−1), we create K2rfree graphs with minimum degree δ and diameter (6r−5)n(2r−1)δ+2r−3+O(1), which are counterexamples to the conjecture for every r>1 and δ>2(r−1)(3r+2)(2r−3)

ErdősHajnaltype results for monotone paths J. Comb. Theory B (IF 1.317) Pub Date : 20210609
János Pach, István TomonAn ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck>0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least nck. This strengthens a result of Bousquet, Lagoutte, and

Many cliques with few edges and bounded maximum degree J. Comb. Theory B (IF 1.317) Pub Date : 20210608
Debsoumya Chakraborti, Da Qi ChenGeneralized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and bounded maximum degree, was recently completely resolved by Chase. Kirsch and Radcliffe raised a natural variant of this problem where the number of edges is fixed instead

Laminar tight cuts in matching covered graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210603
Guantao Chen, Xing Feng, Fuliang Lu, Cláudio L. Lucchesi, Lianzhu ZhangAn edge cut C of a graph G is tight if C∩M=1 for every perfect matching M of G. Barrier cuts and 2separation cuts are called ELPcuts, which are two important types of tight cuts in matching covered graphs. Edmonds, Lovász and Pulleyblank proved that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELPcut. Carvalho, Lucchesi, and Murty made a stronger conjecture:

Rooted topological minors on four vertices J. Comb. Theory B (IF 1.317) Pub Date : 20210604
Koyo Hayashi, Kenichi KawarabayashiFor a graph G and a set Z of four distinct vertices of G, a diamond on Z is a subgraph of G such that, for some labeling Z={v1,v2,v3,v4}, there are three internally disjoint paths P1,P2,P3 with end vertices v1,v2 with v3,v4 on P1,P2, respectively. Therefore, this yields a K4−subdivision with branch vertices on Z. We characterize graphs G that contain no diamond on a prescribed set Z of four vertices

Cubic graphs that cannot be covered with four perfect matchings J. Comb. Theory B (IF 1.317) Pub Date : 20210510
Edita Máčajová, Martin ŠkovieraA conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is 3edgecolourable, the rest of cubic graphs fall into two classes: those that can be covered with four perfect matchings, and those that need at least five. Cubic graphs that require more than four perfect

The structure of clawfree binary matroids J. Comb. Theory B (IF 1.317) Pub Date : 20210423
Peter Nelson, Kazuhiro NomotoA simple binary matroid is called clawfree if none of its rank3 flats are independent sets. These objects can be equivalently defined as the sets E of points in PG(n−1,2) for which E∩P is not a basis of P for any plane P, or as the subsets X of F2n containing no linearly independent triple x,y,z for which x+y,y+z,x+z,x+y+z∉X. We prove a decomposition theorem that exactly determines the structure

Hamiltonian and pseudoHamiltonian cycles and fillings in simplicial complexes J. Comb. Theory B (IF 1.317) Pub Date : 20210423
Rogers Mathew, Ilan Newman, Yuri Rabinovich, Deepak RajendraprasadWe introduce and study a ddimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian ddimensional cycles in Knd (the complete simplicial dcomplex over a vertex set of size n). Hamiltonian dcycles are the simple dcycles of a complete rank, or, equivalently, of size 1+(n−1d). The discussion is restricted to the fields F2 and Q. For d=2, we characterize the n's for which Hamiltonian

On Hamiltonian cycles in hypergraphs with dense link graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210421
Joanna Polcyn, Christian Reiher, Vojtěch Rödl, Bjarne SchülkeWe show that every kuniform hypergraph on n vertices whose minimum (k−2)degree is at least (5/9+o(1))n2/2 contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and SanhuezaMatamala.

An explicit characterization of arctransitive circulants J. Comb. Theory B (IF 1.317) Pub Date : 20210304
Cai Heng Li, Binzhou Xia, Sanming ZhouA reductive characterization of arctransitive circulants was given independently by Kovács in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arctransitive circulants and their automorphism groups. Based on this, we give a proof of the fact that arctransitive circulants are all CIdigraphs.

On the spectral gap and the automorphism group of distanceregular graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210226
Bohdan KivvaWe prove that a distanceregular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group. The minimal degree of a permutation group G is the minimum number of points not fixed by nonidentity elements of G. Lower bounds on the minimal degree

Connectivity and choosability of graphs with no Kt minor J. Comb. Theory B (IF 1.317) Pub Date : 20210218
Sergey Norin, Luke PostleIn 1943, Hadwiger conjectured that every graph with no Kt+1 minor is tcolorable for every t≥0. While Hadwiger's conjecture does not hold for listcoloring, the linear weakening is conjectured to be true. In the 1980 s, Kostochka and Thomason independently proved that every graph with no Kt minor has average degree O(tlogt) and thus is O(tlogt)listcolorable. Recently, the authors and Song proved

A solution of LiXia's problem on sarctransitive solvable Cayley graphs J. Comb. Theory B (IF 1.317) Pub Date : 20210215
JinXin ZhouThis paper gives a solution of Problem 1.8 in [16]. As a corollary, it is shown that every connected nonbipartite Cayley graph on a solvable group of valency at least three is at most 2arctransitive.

Exponentially many 3colorings of planar trianglefree graphs with no short separating cycles J. Comb. Theory B (IF 1.317) Pub Date : 20210208
Carsten ThomassenThe number of proper vertex3colorings of every trianglefree planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2n/17700000. On the other hand, for infinitely many n, there exists a trianglefree planar graph with separating cycles of length 4 and 5 whose number of proper vertex3colorings is <215n/log2(n).

Total weight choosability of graphs: Towards the 123conjecture J. Comb. Theory B (IF 1.317) Pub Date : 20210203
Lu CaoLet G=(V,E) be a graph. A proper total weighting of G is a mapping w:V∪E⟶R such that the following sum for each v∈V:w(v)+∑e∈E(v)w(e) gives a proper vertex colouring of G. For any a,b∈N+, we say that G is total weight (a,b)choosable if for any {Sv:v∈V}⊂[R]a and {Se:v∈E}⊂[R]b, there exists a proper total weighting w of G such that w(v)∈Sv for v∈V and w(e)∈Se for e∈E. A strengthening of the 123 Conjecture

Some extremal results on 4cycles J. Comb. Theory B (IF 1.317) Pub Date : 20210129
Jialin He, Jie Ma, Tianchi YangWe present two extremal results on 4cycles. Let q be a large even integer. First we prove that every (q2+q+1)vertex C4free graph with more than 12q(q+1)2−0.2q edges must be a spanning subgraph of a unique polarity graph. This implies a stability refinement of a special case of the seminal work of Füredi on the extremal number of C4. Second we prove that every (q2+q+1)vertex graph with 12q(q+1)2+1

Obstructions for bounded shrubdepth and rankdepth J. Comb. Theory B (IF 1.317) Pub Date : 20210125
Ojoung Kwon, Rose McCarty, Sangil Oum, Paul WollanShrubdepth and rankdepth are dense analogues of the treedepth of a graph. It is well known that a graph has large treedepth if and only if it has a long path as a subgraph. We prove an analogous statement for shrubdepth and rankdepth, which was conjectured by Hliněný et al. (2016) [11]. Namely, we prove that a graph has large rankdepth if and only if it has a vertexminor isomorphic to a long

Acyclic orientation polynomials and the sink theorem for chromatic symmetric functions J. Comb. Theory B (IF 1.317) Pub Date : 20210122
ByungHak Hwang, WooSeok Jung, KangJu Lee, Jaeseong Oh, SangHoon YuWe define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at −1 up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop “acyclic orientation” analogues of theorems concerning the

The convex dimension of hypergraphs and the hypersimplicial Van KampenFlores Theorem J. Comb. Theory B (IF 1.317) Pub Date : 20210120
Leonardo MartínezSandoval, Arnau PadrolThe convex dimension of a kuniform hypergraph is the smallest dimension d for which there is an injective mapping of its vertices into Rd such that the set of kbarycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete kuniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We