Let be a family of -uniform hypergraphs. The random Turán number is the maximum number of edges in an -free subgraph of , where is the Erdős-Rényi random -graph with parameter . Let denote the -uniform linear cycle of length . For , Mubayi and Yepremyan showed that . This upper bound is not tight when . In this paper, we close the gap for . More precisely, we show that when . Similar results have recently

The Erdős-Sós conjecture states that the maximum number of edges in an -vertex graph without a given -vertex tree is at most . Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a -vertex tree , we construct -vertex connected graphs that are -free with at least

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε>0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G contains δnv(H) copies of H for some δ=δ(ε,H)>0. The current proofs of the removal lemma give only very weak bounds on δ(ε,H), and it is also known that δ(ε,H) is not polynomial in ε unless H is bipartite. Recently

We show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Łuczak, and Thomason from 2004.

A simple graph G with maximum degree Δ is overfull if |E(G)|>Δ⌊|V(G)|/2⌋. The core of G, denoted GΔ, is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals Δ+1 if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with Δ≥3 and Δ(GΔ)≤2, then χ′(G)=Δ+1 implies that G is overfull or G=P⁎, where P⁎ is obtained

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d, d≤4, is at most cdn where c3=1 and c4=9. Whether such a constant cd exists for valencies larger than 4 remains an unanswered question. Further

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph Kn is asymptotically minimised by the random colouring, or equivalently, tH(W)+tH(1−W)≥21−e(H) holds for every graphon W:[0,1]2→[0,1], where tH(.) denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus k or for embeddings into nonorientable surface of genus k are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete

The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G)

We consider two types of matroids defined on the edge set of a graph G: count matroids Mk,ℓ(G), in which independence is defined by a sparsity count involving the parameters k and ℓ, and the C21-cofactor matroid C(G), in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair (k,ℓ), that if G is sufficiently highly connected, then G−e has maximum rank

Graphs of bounded degeneracy are known to contain induced paths of order Ω(log⁡log⁡n) when they contain a path of order n, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to Ω((log⁡n)c) for some constant c>0 depending on the degeneracy. We disprove this conjecture by constructing, for arbitrarily large values of

Abstract not available

We determine the maximum possible number of edges of a graph with n vertices, matching number at most s and clique number at most k for all admissible values of the parameters.

Two subgraphs A,B of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two

In this paper, we prove that every graph with average degree at least s+t+2 has a vertex partition into two parts, such that one part has average degree at least s, and the other part has average degree at least t. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan.

We show that height h posets that have planar cover graphs have dimension O(h6). Previously, the best upper bound was 2O(h3). Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes K5 as a minor.

We prove that every P5-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where Pt denotes the t-vertex path, and for graphs G,H, we say G is H-free if no induced subgraph of G is isomorphic to H). More generally, let us say a class C of graphs is η-bounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree

We prove that for fixed k, every k-uniform hypergraph on n vertices and of minimum codegree at least n/2+o(n) contains every spanning tight k-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions

Alon and Krivelevich proved that for every n-vertex subcubic graph H and every integer q≥2 there exists a (smallest) integer f=f(H,q) such that every Kf-minor contains a subdivision of H in which the length of every subdivision-path is divisible by q. Improving their superexponential bound, we show that f(H,q)≤212qn+8n+14q, which is optimal up to a constant multiplicative factor.

We present three cut trees of graphs, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation R on the vertex set of the graph, which generalizes Gomory-Hu trees. Applying these cut trees, we prove the following: • A pair of vertices {v,w} of a graph G is pendant if λ(v,w)=min⁡{d(v)

We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

Let X be a family of subsets of a finite set E. A matroid on E is called an X-matroid if each set in X is a circuit. We develop techniques for determining when there exists a unique maximal X-matroid in the weak order poset of all X-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of

The Gyárfás-Sumner conjecture says that for every forest H and every integer k, if G is H-free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw,ω)-bounded. The family of (tw,ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection

We say a class C of graphs is clean if for every positive integer t there exists a positive integer w(t) such that every graph in C with treewidth more than w(t) contains an induced subgraph isomorphic to one of the following: the complete graph Kt, the complete bipartite graph Kt,t, a subdivision of the (t×t)-wall or the line graph of a subdivision of the (t×t)-wall. In this paper, we adapt a method

We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor. By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k×k)-wall or the line graph of

We prove that for any ε>0, for any large enough t, there is a graph that admits no Kt-minor but admits a (32−ε)t-colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.

We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result [20] of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete

Let us say a graph is Os-free, where s≥1 is an integer, if there do not exist s cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when s=2, is not well understood. For instance, until now we did not know how to test whether a graph is O2-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that O2-free

Applying a recent extension (2015) of a structure theorem of Robertson, Seymour and Thomas from 1993, in this paper we establish Robertson's magic-tree conjecture from 1997.

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 have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥cn for some constant c>0. In 2014, Babai proved the desired

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999. In this paper we generalise basic ideas from the graph minor series by Robertson

We prove that for every ε>0 there exists n0=n0(ε) such that every regular oriented graph on n>n0 vertices and degree at least (1/4+ε)n has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions

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.