We prove that there exists a constant C so that, for all s,k∈N, if G has average degree at least kCs3 and does not contain Ks,s as a subgraph then it contains an induced subgraph which is C4-free and has average degree at least k. It was known that some function of s and k suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph H, there exists ϵ>0 such that every n-vertex graph with no pivot-minor isomorphic to H contains two sets A,B of vertices such that |A|,|B|⩾ϵn and A is complete or anticomplete to B.

For a finite (not necessarily abelian) group (Γ,⋅), let n(Γ) denote the smallest positive integer n such that for each labelling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich [2] initiated the study of the parameter n(⋅) on cyclic groups and proved n(Zq)=O(qlog⁡q)

Let R be a group and let S be a subset of R. The Haar graph Haar(R,S) of R with connection set S is the graph having vertex set R×{−1,1}, where two distinct vertices (x,−1) and (y,1) are declared to be adjacent if and only if yx−1∈S. The name Haar graph was coined by Tomaž Pisanski in one of the first investigations on this class of graphs.

Let λKnh be the complete h-uniform n-vertex hypergraph in which each edge is repeated λ times. For r:=(r1,…,rk), a (partial)r-factorization of λKnh is a partition of the edges of λKnh into factors F1,…,Fk such that each factor is spanning and the degree of all vertices in each Fi is (at most) ri. Suppose that n≥(h−1)(2m−1). We establish necessary and sufficient conditions that ensure a partial r-factorization

We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids N

For a finite abelian group (Γ,+), let n(Γ) denote the smallest positive integer n such that for each labeling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals 0. Alon and Krivelevich initiated the study of the parameter n(⋅) on cyclic groups and proved that n(Zq)=O(qlog⁡q). Several improvements

Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour (1986) [22] or by a classic result of Chekuri and Chuzhoy (2015) [5], we show that for any natural numbers t and w, and real ε>0, there is an integer W:=W(t,w,ε) such that every graph with treewidth at least W and no

This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments A,B, both with large chromatic number, such that all edges between them are directed from A to B? We describe what we know about this question, and report some progress on several other related

For every k≥2 and Δ, we prove that there exists a constant CΔ,k such that the following holds. For every graph H with χ(H)=k and every tree T with at least CΔ,k|H| vertices and maximum degree at most Δ, the Ramsey number R(T,H) is (k−1)(|T|−1)+σ(H), where σ(H) is the size of a smallest colour class across all proper k-colourings of H. This is tight up to the value of CΔ,k, and confirms a conjecture

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papasoglu and Whyte on quasi-isometries between multi-ended

Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let Pt″ be the complete join of two isolated vertices

Menger's well-known theorem from 1927 characterizes when it is possible to find k vertex-disjoint paths between two sets of vertices in a graph G. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the k paths are required to be pairwise at some distance at least d. The result is known for

Using a probabilistic method, we prove that d(d+1)-connected graphs are rigid in Rd, a conjecture of Lovász and Yemini. Then, using recent results on weakly globally linked pairs, we modify our argument to prove that d(d+1)-connected graphs are globally rigid, too, a conjecture of Connelly, Jordán and Whiteley. The constant d(d+1) is best possible.

A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph G into copies of H1,…,Hm are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with (m+12) edges can be decomposed into subgraphs H1,…,Hm such that each Hi has i edges

An Eulerian orientation is an orientation of the edges of a graph such that every vertex is balanced: its in-degree equals its out-degree. Counting Eulerian orientations corresponds to the crucial partition function in so-called “ice-type models” in statistical physics and is known to be hard for general graphs. For all graphs with good expansion properties and degrees larger than log8⁡n, we derive

Given an r-graph F with r≥2, let ex(n,(t+1)F) denote the maximum number of edges in an n-vertex r-graph with at most t pairwise vertex-disjoint copies of F. Extending several old results and complementing prior work [34] on nondegenerate hypergraphs, we initiate a systematic study on ex(n,(t+1)F) for degenerate hypergraphs F.

Let ex(n,s) denote the maximum number of edges in a triangle-free graph on n vertices which contains no independent sets larger than s. The behaviour of ex(n,s) was first studied by Andrásfai, who conjectured that for s>n/3 this function is determined by appropriately chosen blow-ups of so called Andrásfai graphs. Moreover, he proved ex(n,s)=n2−4ns+5s2 for s/n∈[2/5,1/2] and in earlier work we obtained

The celebrated Kővári-Sós-Turán theorem states that any n-vertex graph containing no copy of the complete bipartite graph Ks,s has at most Os(n2−1/s) edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose

The weak saturation number wsat(n,F) is the minimum number of edges in a graph on n vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of F. In contrast to previous algebraic approaches, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive

We prove that there exists a function f:N→R such that every directed graph G contains either k directed odd cycles where every vertex of G is contained in at most two of them, or a set of at most f(k) vertices meeting all directed odd cycles. We give a polynomial-time algorithm for fixed k which outputs one of the two outcomes. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles

The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on n vertices of diameter greater than two is at least n/C for some universal constant C>0, unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter d≥3 on n vertices is at least Cn/(log⁡n)6 for some

In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈n/r⌉.

For a graph H and an n-vertex graph G, the H-bootstrap process on G is the process which starts with G and, at every time step, adds any missing edges on the vertices of G that complete a copy of H. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum running time (number of time steps before stabilising) of this process over

A pair of graphs (Γ,Σ) is called unstable if their direct product Γ×Σ has automorphisms that do not come from Aut(Γ)×Aut(Σ), and such automorphisms are said to be unexpected. In the special case when Σ=K2, the stability of (Γ,K2) is well studied in the literature, where the so-called two-fold automorphisms of the graph Γ have played an important role. As a generalization of two-fold automorphisms,

We describe a construction of the Tutte polynomial for both matroids and q-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct

We consider k-graphs on n vertices, that is, F⊂([n]k). A k-graph F is called intersecting if F∩F′≠∅ for all F,F′∈F. In the present paper we prove that for k≥7, n≥2k, any intersecting k-graph F with covering number at least three, satisfies |F|≤(n−1k−1)−(n−kk−1)−(n−k−1k−1)+(n−2kk−1)+(n−k−2k−3)+3, the best possible upper bound which was proved in [4] subject to exponential constraints n>n0(k).

An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge pairs of the graph. A classification of orientably-regular embeddings of complete graphs was obtained by L.D. James and G.A. Jones (1985) [10], pointing out interesting

Two families of sets A and B are called cross-t-intersecting if |A∩B|≥t for all A∈A, B∈B. An active problem in extremal set theory is to determine the maximum product of sizes of cross-t-intersecting families. This incorporates the classical Erdős–Ko–Rado (EKR) problem. In the present paper, we prove that if A⊆([n]k) and B⊆([n]k) are cross-t-intersecting with k≥t≥3 and n≥(t+1)(k−t+1), then |A||B|≤(n−tk−t)2

Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an n-vertex 3-uniform linear hypergraph, without a Berge path of length k as a subgraph is at most (k−1)6n for k≥4. The bound is sharp for infinitely many k and n.

We prove that every n-vertex complete simple topological graph generates at least Ω(n) pairwise disjoint 4-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every n-vertex complete simple topological graph drawn in the unit square generates a 4-face with area at most O(1/n). This can be seen as a topological variant of the Heilbronn problem for quadrilaterals.

The Grassmann graph Jq(n,D) is a graph on the D-dimensional subspaces of Fqn with two subspaces being adjacent if their intersection has dimension D−1. Characterizing these graphs by their intersection numbers is an important step towards a solution of the classification problem for (PandQ)-polynomial association schemes, posed by Bannai and Ito in their monograph “Algebraic Combinatorics I” (1984)

We investigate the minimum number of cycles of specified lengths in planar n-vertex triangulations G. We prove that this number is Ω(n) for any cycle length at most 3+max⁡{rad(G⁎),⌈(n−32)log3⁡2⌉}, where rad(G⁎) denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian n-vertex triangulations

Sumner's universal tournament conjecture states that every (2n−2)-vertex tournament should contain a copy of every n-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by Häggkvist and Thomason (for number of leaves) and Kühn, Mycroft and Osthus (for maximum

For r,n⩾2, an ordered r-uniform matching of size n is an r-uniform hypergraph on a linearly ordered vertex set V, with |V|=rn, consisting of n pairwise disjoint edges. There are 12(2rr) different ways two edges may intertwine, called here patterns. Among them we identify 3r−1 collectable patterns P, which have the potential of appearing in arbitrarily large quantities called P-cliques.

If G is a graph, A and B its induced subgraphs, and f:A→B an isomorphism, we say that f is a partial automorphism of G. In 1992, Hrushovski proved that graphs have the extension property for partial automorphisms (EPPA, also called the Hrushovski property), that is, for every finite graph G there is a finite graph H, an EPPA-witness for G, such that G is an induced subgraph of H and every partial automorphism

We study the generic volume-rigidity of (d−1)-dimensional simplicial complexes in Rd−1, and show that the volume-rigidity of a complex can be identified in terms of its exterior shifting. In addition, we establish the volume-rigidity of triangulations of several 2-dimensional surfaces and prove that, in all dimensions >1, volume-rigidity is not characterized by a corresponding hypergraph sparsity property

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs H with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect F-tilings (for an arbitrary fixed graph F) by replacing the F-tiling with the aforementioned graphs

A graph U is universal for a graph class C∋U, if every G∈C is a minor of U. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding K5, or K3,3, or K∞ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that do not have a universal element.

For a given graph H, its subdivisions carry the same topological structure. The existence of H-subdivisions within a graph G has deep connections with topological, structural and extremal properties of G. One prominent example of such a connection, due to Bollobás and Thomason and independently Komlós and Szemerédi, asserts that the average degree of G being d ensures a KΩ(d)-subdivision in G. Although

Pastures are a class of field-like algebraic objects which include both partial fields and hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results

An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. We prove that every locally finite quasi-transitive graph avoiding a minor has a tree-decomposition

Graphings serve as limit objects for bounded-degree graphs. We define the “cycle matroid” of a graphing as a submodular setfunction, with values in , which generalizes (up to normalization) the cycle matroid of finite graphs. We prove that for a Benjamini–Schramm convergent sequence of graphs, the total rank, normalized by the number of nodes, converges to the total rank of the limit graphing.

Let and be hypergraphs on vertices, and suppose has large enough minimum degree to necessarily contain a copy of as a subgraph. We give a general method to randomly embed into with good “spread”. More precisely, for a wide class of , we find a randomised embedding with the following property: for every , for any partial embedding of vertices of into , the probability that extends is at most . This

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph

Let be a graph attaining the maximum spectral radius among all connected nonregular graphs of order with maximum degree Δ. Let be the spectral radius of . A nice conjecture due to Liu et al. (2007) asserts that for each fixed Δ. Concerning an important structural property of the extremal graphs , Liu and Li (2008) put forward another conjecture which states that has exactly one vertex of degree strictly

Given a finite simple graph Γ on vertices its complementary prism is the graph that is obtained from Γ and its complement by adding a perfect matching where each its edge connects two copies of the same vertex in Γ and . It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of is described for an arbitrary graph Γ. In particular, it is shown that the ratio

Let be an -vertex graph. The vertex arboricity of is the least integer such that can be partitioned into parts and each part induces a forest in . We show that for sufficiently large , every -vertex graph with and contains an -factor, where or . The result can be viewed an analogue of the Alon–Yuster theorem in Ramsey–Turán theory, which generalizes the results of Balogh–Molla–Sharifzadeh and Knierim–Su

A graph is called if for every induced subgraph of , where is the clique number of and its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph is perfect if and only if its complement is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.

In this note, we prove that finite CAT(0) cube complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). This result was conjectured by Haslegrave, Scott, Tamitegama, and Tan (2023). The reconstruction of a finite cell complex from the boundary distances is the discrete version of the boundary rigidity problem, which is a classical problem from Riemannian geometry

In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most if it has a fractional coloring in which each vertex receives a subset of of measure at least . We introduce and develop the theory of “fractional colorings with local demands” wherein each vertex “demands”

A -coloring of is an edge-coloring of where every 4-clique spans at least five colors. We show that there exist -colorings of using colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process

The study of graph discrepancy problems, initiated by Erdős in the 1960s, has received renewed attention in recent years. In general, given a 2-edge-coloured graph , one is interested in embedding a copy of a graph in with large discrepancy (i.e. the copy of contains significantly more than half of its edges in one colour).

We prove that for any graph of maximum degree at most Δ, the zeros of its chromatic polynomial (in ) lie inside the disc of radius 5.94Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ.

A graph is -linked if, for any distinct vertices in , there exist disjoint connected subgraphs of such that and . A fundamental result in structural graph theory is the characterization of -linked graphs. It appears to be difficult to characterize -linked graphs for . In this paper, we provide a partial characterization of -linked graphs. This implies that every -connected graphs is -linked and for

Given a graph , consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of . For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted

Let denote an -uniform hypergraph with edges and vertices, where (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are for all , and for . We prove that as . In the case , we prove as , and for all .

Given a graph , let denote the smallest for which the following holds. We can assign a -colouring of the edge set of to each vertex in with the property that for any copy of in , there is some such that every edge in has a different colour in .

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs of a graph of the neighborhood set system of is sandwiched between and , where denotes the degeneracy of . We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization