• Canonical trees of tree-decompositions
J. Comb. Theory B (IF 1.317) Pub Date : 2021-09-14
Johannes Carmesin, Matthias Hamann, Babak Miraftab

We prove that every graph has a canonical tree of tree-decompositions that distinguishes all principal tangles (these include the ends and various kinds of large finite dense structures) efficiently. Here ‘trees of tree-decompositions’ are a slightly weaker notion than ‘tree-decompositions’ but much more well-behaved than ‘tree-like 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 : 2021-08-31
S. Das, A. Pokrovskiy, B. Sudakov

Graph 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

• N-detachable pairs in 3-connected matroids III: The theorem
J. Comb. Theory B (IF 1.317) Pub Date : 2021-08-27
Nick Brettell, Geoff Whittle, Alan Williams

Let M be a 3-connected matroid, and let N be a 3-connected minor of M. A pair {x1,x2}⊆E(M) is N-detachable if one of the matroids M/x1/x2 or M﹨x1﹨x2 is 3-connected and has an N-minor. 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 N-detachable pair after possibly performing a single Δ-Y or Y-Δ exchange, or M is essentially N with a spike

• On the spectrum of Hamiltonian cycles in the n-cube
J. Comb. Theory B (IF 1.317) Pub Date : 2021-08-27
A.L. Perezhogin

The spectrum of a Hamiltonian cycle (Gray code) in a Boolean n-cube 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

• Erdős-Hajnal for cap-free graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-08-06
Maria Chudnovsky, Paul Seymour

A “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 |G|ε; this is

• A proof of the upper matching conjecture for large graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-08-05
Ewan Davies, Matthew Jenssen, Will Perkins

We 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 d-regular bipartite graph maximizes the number of independent sets and matchings

• The inverse Kazhdan-Lusztig polynomial of a matroid
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-30
Alice L.L. Gao, Matthew H.Y. Xie

In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley 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 : 2021-07-26
Bohdan Kivva

One of the central results in the representation theory of distance-regular graphs classifies distance-regular 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 distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism

• The evolution of the structure of ABC-minimal trees
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-24

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor that found diverse chemical applications. Characterizing trees with minimum ABC-index 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

• Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-24
Zdeněk Dvořák, Daniel Král', Robin Thomas

We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring 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 : 2021-07-14
Matija Bucić, Nemanja Draganić, Benny Sudakov, Tuan Tran

The 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 r-graphs 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 r-graph that can not be avoided in any r-graph on n

• Enlarging vertex-flames in countable digraphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-13
Joshua Erde, J. Pascal Gollin, Attila Joó

A rooted digraph is a vertex-flame 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 vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root

• On edge-primitive 3-arc-transitive graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-13
Michael Giudici, Carlisle S.H. King

This paper begins the classification of all edge-primitive 3-arc-transitive 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 vertex-stabiliser acting faithfully on the set of neighbours.

• On the cop number of toroidal graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-07
Florian Lehner

We 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.

• Edge-critical subgraphs of Schrijver graphs II: The general case
J. Comb. Theory B (IF 1.317) Pub Date : 2021-07-03
Tomáš Kaiser, Matěj Stehlík

We give a simple combinatorial description of an (n−2k+2)-chromatic edge-critical subgraph of the Schrijver graph SG(n,k), itself an induced vertex-critical subgraph of the Kneser graph KG(n,k). This extends the main result of Kaiser and Stehlík (2020)  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 : 2021-06-30

For 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 : 2021-06-29
Jan Kurkofka

We show that, up to minor-equivalence, the Farey graph is the unique minor-minimal graph that is infinitely edge-connected but such that every two vertices can be finitely separated.

• Packing and covering immersions in 4-edge-connected graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-25
Chun-Hung Liu

A 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 edge-variant of the Erdős-Pósa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph H, there exists a function f such that for every 4-edge-connected graph

• Polynomial bounds for centered colorings on proper minor-closed graph classes
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-23
Michał Pilipczuk, Sebastian Siebertz

For p∈N, a coloring λ of the vertices of a graph G is p-centered 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 , , 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 : 2021-06-23
Tom Kelly, Luke Postle

We correct an error that appears in Kelly and Postle (2020) . 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 : 2021-06-21
François Dross, Frédéric Havet

A digraph is n-unavoidable 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 χp-bounded classes
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-16
Yiting Jiang, Jaroslav Nešetřil, Patrice Ossona de Mendez

χ-bounded classes are studied here in the context of star colorings and, more generally, χp-colorings. 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. χp-boundedness

• On the intersection conjecture for infinite trees of matroids
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-15
Nathan Bowler, Johannes Carmesin

Using 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 2-separations into finite parts.

• Counterexamples to a conjecture of Erdős, Pach, Pollack and Tuza
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-11
Éva Czabarka, Inne Singgih, László A. Székely

Erdős et al. (1989)  conjectured that the diameter of a K2r-free 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 K2r-free 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)

• Erdős-Hajnal-type results for monotone paths
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-09
János Pach, István Tomon

An 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 : 2021-06-08
Debsoumya Chakraborti, Da Qi Chen

Generalized 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 : 2021-06-03
Guantao Chen, Xing Feng, Fuliang Lu, Cláudio L. Lucchesi, Lianzhu Zhang

An edge cut C of a graph G is tight if |C∩M|=1 for every perfect matching M of G. Barrier cuts and 2-separation cuts are called ELP-cuts, 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 ELP-cut. Carvalho, Lucchesi, and Murty made a stronger conjecture:

• Rooted topological minors on four vertices
J. Comb. Theory B (IF 1.317) Pub Date : 2021-06-04
Koyo Hayashi, Ken-ichi Kawarabayashi

For 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 : 2021-05-10
Edita Máčajová, Martin Škoviera

A 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 3-edge-colourable, 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 claw-free binary matroids
J. Comb. Theory B (IF 1.317) Pub Date : 2021-04-23
Peter Nelson, Kazuhiro Nomoto

A simple binary matroid is called claw-free if none of its rank-3 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 pseudo-Hamiltonian cycles and fillings in simplicial complexes
J. Comb. Theory B (IF 1.317) Pub Date : 2021-04-23
Rogers Mathew, Ilan Newman, Yuri Rabinovich, Deepak Rajendraprasad

We introduce and study a d-dimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian d-dimensional cycles in Knd (the complete simplicial d-complex over a vertex set of size n). Hamiltonian d-cycles are the simple d-cycles 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 : 2021-04-21
Joanna Polcyn, Christian Reiher, Vojtěch Rödl, Bjarne Schülke

We show that every k-uniform 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 Sanhueza-Matamala.

• An explicit characterization of arc-transitive circulants
J. Comb. Theory B (IF 1.317) Pub Date : 2021-03-04
Cai Heng Li, Binzhou Xia, Sanming Zhou

A reductive characterization of arc-transitive circulants was given independently by Kovács in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arc-transitive circulants and their automorphism groups. Based on this, we give a proof of the fact that arc-transitive circulants are all CI-digraphs.

• On the spectral gap and the automorphism group of distance-regular graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-02-26
Bohdan Kivva

We prove that a distance-regular 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 non-identity 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 : 2021-02-18
Sergey Norin, Luke Postle

In 1943, Hadwiger conjectured that every graph with no Kt+1 minor is t-colorable for every t≥0. While Hadwiger's conjecture does not hold for list-coloring, 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(tlog⁡t) and thus is O(tlog⁡t)-list-colorable. Recently, the authors and Song proved

• A solution of Li-Xia's problem on s-arc-transitive solvable Cayley graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-02-15
Jin-Xin Zhou

This paper gives a solution of Problem 1.8 in . As a corollary, it is shown that every connected non-bipartite Cayley graph on a solvable group of valency at least three is at most 2-arc-transitive.

• Exponentially many 3-colorings of planar triangle-free graphs with no short separating cycles
J. Comb. Theory B (IF 1.317) Pub Date : 2021-02-08
Carsten Thomassen

The number of proper vertex-3-colorings of every triangle-free 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 triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is <215n/log2⁡(n).

• Total weight choosability of graphs: Towards the 1-2-3-conjecture
J. Comb. Theory B (IF 1.317) Pub Date : 2021-02-03
Lu Cao

Let 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 1-2-3 Conjecture

• Some extremal results on 4-cycles
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-29
Jialin He, Jie Ma, Tianchi Yang

We present two extremal results on 4-cycles. Let q be a large even integer. First we prove that every (q2+q+1)-vertex C4-free 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 shrub-depth and rank-depth
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-25
O-joung Kwon, Rose McCarty, Sang-il Oum, Paul Wollan

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný et al. (2016) . Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long

• Acyclic orientation polynomials and the sink theorem for chromatic symmetric functions
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-22
Byung-Hak Hwang, Woo-Seok Jung, Kang-Ju Lee, Jaeseong Oh, Sang-Hoon Yu

We 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 Kampen-Flores Theorem
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-20

The convex dimension of a k-uniform hypergraph is the smallest dimension d for which there is an injective mapping of its vertices into Rd such that the set of k-barycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete k-uniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We

• A Cantor-Bernstein-type theorem for spanning trees in infinite graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-19
Joshua Erde, J. Pascal Gollin, Atilla Joó, Paul Knappe, Max Pitz

We show that if a graph admits a packing and a covering both consisting of λ many spanning trees, where λ is some infinite cardinal, then the graph also admits a decomposition into λ many spanning trees. For finite λ the analogous question remains open, however, a slightly weaker statement is proved.

• A Harary-Sachs theorem for hypergraphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-14
Gregory J. Clark, Joshua N. Cooper

We generalize the Harary-Sachs theorem to k-uniform hypergraphs: the codegree-d coefficient of the characteristic polynomial of a uniform hypergraph H can be expressed as a weighted sum of subgraph counts over certain multi-hypergraphs with d edges. We include a detailed description of the aforementioned multi-hypergraphs and a formula for their corresponding weights.

• Long cycles, heavy cycles and cycle decompositions in digraphs
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-11
Charlotte Knierim, Maxime Larcher, Anders Martinsson, Andreas Noever

Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most ⌊(n−1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) , Dean (1986) , and Bollobás and Scott (1996)  it was analogously conjectured that every directed Eulerian graph can be decomposed

• On the rational Turán exponents conjecture
J. Comb. Theory B (IF 1.317) Pub Date : 2021-01-12
Dong Yeap Kang, Jaehoon Kim, Hong Liu

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n,F)=Θ(nr). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 0,1,75,2, and the numbers

• Regular Cayley maps for dihedral groups
J. Comb. Theory B (IF 1.317) Pub Date : 2020-12-29
István Kovács, Young Soo Kwon

An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut∘M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut∘M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified

• The feasible region of hypergraphs
J. Comb. Theory B (IF 1.317) Pub Date : 2020-12-24
Xizhi Liu, Dhruv Mubayi

Let F be a family of r-uniform hypergraphs. The feasible region Ω(F) of F is the set of points (x,y) in the unit square such that there exists a sequence of F-free r-uniform hypergraphs whose shadow density approaches x and whose edge density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x,y)∈Ω(F) is the Turán density π(F), and

• Kr-factors in graphs with low independence number
J. Comb. Theory B (IF 1.317) Pub Date : 2020-12-24
Charlotte Knierim, Pascal Su

A classical result by Hajnal and Szemerédi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a Kr-factor. Namely, any graph on n vertices, with minimum degree δ(G)≥(1−1r)n and r dividing n has a Kr-factor. This result is tight but the extremal examples are unique in that they all have a large independent set which is the bottleneck. Nenadov and Pehova

• A Menger-like property of tree-cut width
J. Comb. Theory B (IF 1.317) Pub Date : 2020-12-22
Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, Dimitrios M. Thilikos

In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan (2015) as a possible edge-version of tree decompositions. We show

• Tetravalent half-arc-transitive graphs with unbounded nonabelian vertex stabilizers
J. Comb. Theory B (IF 1.317) Pub Date : 2020-12-16
Binzhou Xia

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian

• Graph functionality
J. Comb. Theory B (IF 1.317) Pub Date : 2020-11-17
Bogdan Alecu, Aistis Atminas, Vadim Lozin

In the present paper, we introduce the notion of graph functionality, which generalises simultaneously several other graph parameters, such as degeneracy or clique-width, in the sense that bounded degeneracy or bounded clique-width imply bounded functionality. Moreover, we show that this generalisation is proper by revealing classes of graphs of unbounded degeneracy and clique-width, where functionality

• Reconfiguring colorings of graphs with bounded maximum average degree
J. Comb. Theory B (IF 1.317) Pub Date : 2020-11-12
Carl Feghali

The reconfiguration graph Rk(G) for the k-colorings of a graph G has as vertex set the set of all possible k-colorings of G and two colorings are adjacent if they differ in the color of exactly one vertex of G. Let d,k≥1 be integers such that k≥d+1. We prove that for every ϵ>0 and every graph G with n vertices and maximum average degree d−ϵ, Rk(G) has diameter O(n(log⁡n)d−1). This significantly strengthens

• Sparse hypergraphs: New bounds and constructions
J. Comb. Theory B (IF 1.317) Pub Date : 2020-11-05
Gennian Ge, Chong Shangguan

Let fr(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v+1 vertices. The study of fr(n,v,e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known. Conjecture: nk−o(1)k≥2 and e≥3 as n→∞. For r=3,e=3,k=2, the bound n2−o(1)

• Dismantlability, connectedness, and mixing in relational structures
J. Comb. Theory B (IF 1.317) Pub Date : 2020-10-22
Raimundo Briceño, Andrei Bulatov, Víctor Dalmau, Benoît Larose

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have been shown to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved—like, for example, dismantlability—and

• Stability of graph pairs
J. Comb. Theory B (IF 1.317) Pub Date : 2020-10-27
Yan-Li Qin, Binzhou Xia, Jin-Xin Zhou, Sanming Zhou

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs (Γ,Σ) is stable if Aut(Γ×Σ)≅Aut(Γ)×Aut(Σ) and unstable otherwise, where Γ×Σ is the direct product of Γ and Σ. An unstable graph pair (Γ,Σ) is said to be a nontrivially unstable graph pair if Γ and Σ are connected coprime graphs, at least

• 3-List-coloring graphs of girth at least five on surfaces
J. Comb. Theory B (IF 1.317) Pub Date : 2020-10-13
Luke Postle

Grötzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any

• A unified view of inequalities for distance-regular graphs, part I
J. Comb. Theory B (IF 1.317) Pub Date : 2020-10-05
Arnold Neumaier, Safet Penjić

In this paper, we introduce the language of a configuration and of t-point counts for distance-regular graphs (DRGs). Every t-point count can be written as a sum of (t−1)-point counts. This leads to a system of linear equations and inequalities for the t-point counts in terms of the intersection numbers, i.e., a linear constraint satisfaction problem (CSP). This language is a very useful tool for a

• On some topological and combinatorial lower bounds on the chromatic number of Kneser type hypergraphs
J. Comb. Theory B (IF 1.317) Pub Date : 2020-10-01
Soheil Azarpendar, Amir Jafari

In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers n,k,r,s with n≥rk and 2≤s≤r, the r-uniform general Kneser hypergraph KGsr(n,k), has all k-subsets of {1,…,n} as the vertex set and all multi-sets {A1,…,Ar} of k-subsets with s-wise empty intersections as the edge set. The case r=s=2, was considered by Kneser

• A counterexample to prism-hamiltonicity of 3-connected planar graphs
J. Comb. Theory B (IF 1.317) Pub Date : 2020-09-29
Simon Špacapan

The prism over a graph G is the Cartesian product of G with the complete graph K2. A graph G is hamiltonian if there exists a spanning cycle in G, and G is prism-hamiltonian if the prism over G is hamiltonian. Rosenfeld and Barnette (1973)  conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.

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