In this paper, we prove that for any k≥3, there exist infinitely many minimal asymmetric k-uniform hypergraphs. This is in a striking contrast to k=2, where it has been proved recently that there are exactly 18 minimal asymmetric graphs. We also determine, for every k≥1, the minimum size of an asymmetric k-uniform hypergraph.

The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G). Here we show that this statement does not remain true for systems of

A family of subsets of [n] is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for n≥2k+ckln⁡k, almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for n≥2k+100ln⁡k

A theorem of Mader shows that every graph with average degree at least eight has a K6 minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have K6 minors, but minimum degree six is certainly not enough. For every ε>0 there are arbitrarily

Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a Kt-minor. Holroyd [11] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G, then G contains a Kt-minor rooted at S. We prove this conjecture in the first open case of t=4. Notably, our result also directly implies a stronger

Robertson and Seymour constructed for every graph G a tree-decomposition that efficiently distinguishes all the tangles in G. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical. The key ingredient is an axiomatisation of ‘local properties’ of tangles. Generalisations to locally finite

Let M be an excluded minor for the class of P-representable matroids for some partial field P, let N be a 3-connected strong P-stabilizer that is non-binary, and suppose M has a pair of elements {a,b} such that M﹨a,b is 3-connected with an N-minor. Suppose also that |E(M)|≥|E(N)|+11 and M﹨a,b is not N-fragile. In the prequel to this paper, we proved that M﹨a,b is at most five elements away from an

We show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be determined from the pairwise graph distances between boundary vertices. A similar result holds for disc quadrangulations with all internal vertices having degree at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rödl showed in the 1980s that every H-free graph has large parts that are very sparse or very dense. More precisely, let us say that a graph F on n vertices is ε-restricted if either F or its complement has maximum degree at most εn. Rödl proved that for every graph H, and every ε>0, every H-free graph

The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of matroids representable over the Hydra-5 partial field, and the 3-connected matroids in the class with a U2,5- or U3,5-minor are precisely those with six inequivalent

In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the equivalence class of a tree under the embeddability relation is, up to isomorphism, either trivial or infinite. In 2022 Abdi, et al. provided a rigorous exposition of a counter-example to TAC developed by Tetano in his 2008 PhD thesis. In this paper we provide a positive answer to TAC for a weaker type of graph relation: the

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sós showing that any triangle-intersecting family of graphs on n vertices has size at most 2(n2)−3, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to Kt-intersecting families. We prove these conjectures for t∈{3,4}, showing

Let k and n be two integers, with k≥3, n≡0(modk), and n sufficiently large. We determine the (k−1)-degree threshold for the existence of a rainbow perfect matchings in n-vertex k-uniform hypergraph. This implies the result of Rödl, Ruciński, and Szemerédi on the (k−1)-degree threshold for the existence of perfect matchings in n-vertex k-uniform hypergraphs. In our proof, we identify the extremal configurations

In this paper, we determine the class of finite 2-arc-transitive bicirculants. We show that a connected 2-arc-transitive bicirculant is one of the following graphs: C2n where n⩾2, K2n where n⩾2, Kn,n where n⩾3, Kn,n−nK2 where n⩾4, B(PG(d−1,q)) and B′(PG(d−1,q)) where d≥3 and q is a prime power, X1(4,q) where q≡3(mod4) is a prime power, Kq+12d where q is an odd prime power and d≥2 dividing q−1, ATQ(1+q

A well-known result of Verstraëte [23] shows that for each integer k≥2 every graph G with average degree at least 8k contains cycles of k consecutive even lengths, the shortest of which is of length at most twice the radius of G. We establish two extensions of Verstraëte's result for linear cycles in linear r-uniform hypergraphs. We show that for any fixed integers r≥3 and k≥2, there exist constants

Let G=(V,E) be a finite undirected graph. Orient the edges of G in an arbitrary way. A 2-cycle on G is a function d:E2→Z such for each edge e, d(e,⋅) and d(⋅,e) are circulations on G, and d(e,f)=0 whenever e and f have a common vertex. We show that each 2-cycle is a sum of three special types of 2-cycles: cycle-pair 2-cycles, Kuratowski 2-cycles, and quad 2-cycles. In the case that the graph is Kuratowski

We propose local versions of Hadwiger's Conjecture, where only balls of radius Ω(log⁡(v(G))) around each vertex are required to be Kt-minor-free. We ask: if a graph is locally-Kt-minor-free, is it t-colourable? We show that the answer is yes when t≤5, even in the stronger setting of list-colouring, and we complement this result with a O(log⁡v(G))-round distributed colouring algorithm in the LOCAL model

Let F1,…,Fℓ be families of subsets of {1,…,n}. Suppose that for distinct k,k′ and arbitrary F1∈Fk,F2∈Fk′ we have |F1∩F2|⩽m. What is the maximal value of |F1|…|Fℓ|? In this work we find the asymptotic of this product as n tends to infinity for constant ℓ and m. This question is related to a conjecture of Bohn et al. that arose in the 2-level polytope theory and asked for the largest product of the number

We find the exact value of the Ramsey number R(C2ℓ,K1,n), when ℓ and n=O(ℓ10/9) are large. Our result is closely related to the behaviour of Turán number ex(N,C2ℓ) for an even cycle whose length grows quickly with N.

We study relation between two interpretations of the Tutte polynomial of a matroid perspective M1→M2 on a set E given with a linear ordering <. A well known interpretation uses internal and external activities on a family B(M1,M2) of the sets independent in M1 and spanning in M2. Recently we introduced another interpretation based on a family D(M1,M2;<) of “cyclic bases” of M1→M2 with respect to <

Bland and Las Vergnas proved that orientations of binary matroids are induced by totally unimodular representations. (A related result is due to Minty.) Lee and Scobee proved that orientations of ternary matroids are induced by dyadic representations. In this paper we prove that consistently ordered orientations of quaternary matroids are induced by golden-mean representations.

For any positive integer t, a t-broom is a graph obtained from K1,t+1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t-brooms, we have χ(G)=o(ω(G)t+1), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. When t=2, this answers a question of Schiermeyer and Randerath. Moreover, for t=2, we strengthen the bound on χ(G) to 7ω(G)2,

A graph H is ubiquitous if every graph G that for every natural number n contains n vertex-disjoint H-minors contains infinitely many vertex-disjoint H-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.

Dujmović et al. [J. ACM '20] proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, p-centered colouring, and adjacency labelling. This paper

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree much more than the number of atoms in the universe (δ(G)≥101010), then it contains a pillar, which is a graph that consists of two vertex-disjoint cycles of the same length, s say, along with s vertex-disjoint paths of the same length3 which connect matching vertices in order around the cycles. Despite the simplicity of the structure

This paper proves the following result: If G is a planar graph and L is a 4-list assignment of G such that |L(x)∩L(y)|≤2 for every edge xy, then G is L-colourable. This answers a question asked by Kratochvíl et al. (1998) [10].

A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least Ck2 has a subdivision of Kk, the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a Kk-subdivision

For integer n>0, let f(n) be the number of rows of the largest all-0 or all-1 square submatrix of M, minimized over all n×n 0/1-matrices M. Thus f(n)=O(log⁡n). But let us fix a matrix H, and define fH(n) to be the same, minimized over all n×n 0/1-matrices M such that neither M nor its complement (that is, change all 0's to 1's and vice versa) contains H as a submatrix. It is known that fH(n)≥εnc, where

Grosu (2016) [11] asked if there exist an integer r≥3 and a finite family of r-graphs whose Turán density, as a real number, has (algebraic) degree greater than r−1. In this note we show that, for all integers r≥3 and d, there exists a finite family of r-graphs whose Turán density has degree at least d, thus answering Grosu's question in a strong form.

We prove that for every t∈N there is a constant γt such that every graph with twin-width at most t and clique number ω has chromatic number bounded by 2γtlog4t+3⁡ω. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially χ-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially χ-bounded

A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In an earlier paper [1], Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: given μ>0 and integers ℓ,r and n with n∈rN, is it true that there exists an α>0 such that every n-vertex graph G with δ(G)≥max⁡{12,r−ℓr}n+μn and αℓ(G)≤αn contains a Kr-factor? We give a negative answer to this question for the case ℓ≥3r4 by giving a family of constructions using the so-called cover thresholds

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3⋅4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph)

Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G∖S belongs to G. We denote by Ak(G) the set of all graphs that are k-apices of G. We prove that every graph in the obstruction set of Ak(G), i.e., the minor-minimal set of graphs not belonging to Ak(G), has order at most 2222poly(k), where poly is a polynomial function

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs (G,γ) where γ assigns to each edge of an undirected graph G an element of an abelian group Γ. As a consequence, we prove that Γ-nonzero cycles (cycles whose edge labels sum to a non-identity element of Γ) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that

We present a systematic investigation into how tree-decompositions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be distinguished, and characterise which subsets of ends can be displayed by a tree-decomposition of finite adhesion. In particular, we show that a subset Ψ of the ends

In this paper we investigate the bipartite analogue of the strong Erdős-Hajnal property. We prove that for every forest H and every τ with 0<τ≤1, there exists ε>0, such that if G has a bipartition (A,B) and does not contain H as an induced subgraph, and has at most (1−τ)|A|⋅|B| edges, then there is a stable set X of G with |X∩A|≥ε|A| and |X∩B|≥ε|B|. No graphs H except forests have this property.

Let G=(V,E) be a simple graph with n vertices and m edges, P(G,k) be the chromatic polynomial of G, and P(G,L) be the number of L-colorings of G for any k-assignment L. In this article, we show that when k≥m−1≥3, P(G,L)−P(G,k) is bounded below by ((k−m+1)kn−3+(k−m+3)c3kn−5)∑uv∈E|L(u)∖L(v)|, where c≥(m−1)(m−3)8, and in particular, if G is K3-free, then c≥(m−22)+2m−3. Consequently, P(G,L)≥P(G,k) whenever

The proper orientation number χ→(G) of a graph G is the minimum k such that there exists an orientation of the edges of G with all vertex-outdegrees at most k and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper orientation number are resolved. First it is shown, that χ→(G) of any planar graph G is at most 14. Secondly, it is shown that for every

By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has chromatic number at most (C+o(1))Δ/log⁡Δ for some universal constant C>0. Using the entropy compression method, Molloy proved that one can in fact take C=1. Here we show that for every q⩾(1+o(1))Δ/log⁡Δ, the number c(G,q) of proper q-colorings of G satisfiesc(G,q)⩾(1−1q)m((1−o(1))q)n, where n=|V(G)| and m=|E(G)|. Except

We investigate the question how ‘small’ a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph H containing all elements of a graph class G. These conditions imply that such a graph H exists for the class Gd consisting of

We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.

We describe an infinite family of strongly 2-connected oriented graphs (that is, directed graphs with no multiple arcs) containing no arc whose reversal decreases the number of directed cycles.

For any small positive real ε and integer t>1ε, we build a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2(1−ε)t. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least 2−ε. Except for the one in oriented twin-width

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1−ℓ many 1s and at most m+ℓ many 1s, i.e., the vertices in the middle 2ℓ levels, has a Hamilton cycle for any m≥1 and 1≤ℓ≤m+1. This problem was raised independently by Buck and Wiedemann, Savage, Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization

Consider two independent Erdős–Rényi G(N,1/2) graphs. We show that with probability tending to 1 as N→∞, the largest induced isomorphic subgraph has size either ⌊xN−εN⌋ or ⌊xN+εN⌋, where xN=4log2⁡N−2log2⁡log2⁡N−2log2⁡(4/e)+1 and εN=(4log2⁡N)−1/2. Using similar techniques, we also show that if Γ1 and Γ2 are independent G(n,1/2) and G(N,1/2) random graphs, then Γ2 contains an isomorphic copy of Γ1 as

It is known that A-paths of length 0 mod m satisfy the Erdős-Pósa property if m=2 or m=4, but not if m>4 is composite. We show that if p is prime, then A-paths of length 0 mod p satisfy the Erdős-Pósa property. More generally, in the framework of undirected group-labeled graphs, we characterize the abelian groups Γ and elements ℓ∈Γ for which the Erdős-Pósa property holds for A-paths of weight ℓ.

Given k≥2 and two k-graphs (k-uniform hypergraphs) F and H, an F-factor in H is a set of vertex disjoint copies of F that together covers the vertex set of H. Lenz and Mubayi studied the F-factor problems in quasi-random k-graphs with minimum degree Ω(nk−1). In particular, they constructed a sequence of 1/8-dense quasi-random 3-graphs H(n) with minimum degree Ω(n2) and minimum codegree Ω(n) but with

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called edge-ordered if its edges are linearly ordered. This notion allows us to study graphs (and in particular their maximum number of edges) when a subgraph is forbidden with a specific edge-order but the same underlying graph may appear with a different edge-order. We prove an Erdős-Stone-Simonovits-type

Let G be a bridgeless cubic graph. The Berge–Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan–Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs

Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field F, the list needs to contain only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J

Abstract not available

In this paper, all graphs are assumed to be finite. For s≥1 and a graph Γ, if for every pair of isomorphic connected induced subgraphs on at most s vertices there exists an automorphism of Γ mapping the first to the second, then we say that Γ is s-connected-set-homogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at most s vertices can be extended to an automorphism

Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three

In this paper we investigate the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph G and the maximal number of distinct degrees appearing in an induced subgraph of G, denoted respectively by hom⁡(G) and f(G). Our main theorem improves estimates due to several earlier researchers and shows that if G is an n-vertex graph with hom⁡(G)≥n1/2

Let G be a Kp-free graph. We say e is a Kp-saturating edge of G if e∉E(G) and G+e contains a copy of Kp. Denote by fp(n,m) the minimum number of Kp-saturating edges that an n-vertex Kp-free graph with m edges can have. Erdős and Tuza conjectured that f4(n,⌊n2/4⌋+1)=(1+o(1))n216. Balogh and Liu disproved this by showing f4(n,⌊n2/4⌋+1)=(1+o(1))2n233. They believed that a natural generalization of their

Lovász and Cherkassky discovered in the 1970s independently that if G is a finite graph with a given set T of terminal vertices such that G is inner Eulerian with respect to T, then the maximal number of edge-disjoint paths connecting distinct vertices in T is ∑t∈Tλ(t,T−t) where λ is the local edge-connectivity function. The optimality of a system of edge-disjoint T-paths in the Lovász-Cherkassky theorem

For a simple graph F, let Ex(n,F) and Exsp(n,F) denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an n-vertex graph without any copy of the graph F, respectively. The Turán graph Tn,r is the complete r-partite graph on n vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with

The chromatic index χ′(G) of a graph G is the least number of colors assigned to the edges of G such that no two adjacent edges receive the same color. The total chromatic number χ″(G) of a graph G is the least number of colors assigned to the edges and vertices of G such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color