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

A graph is for a -tuple of graphs if for every -coloring of the edges of there exists a monochromatic copy of in color for some . Over the last few decades, researchers have investigated a number of questions related to this notion, aiming to understand the properties of graphs that are -Ramsey for a fixed tuple. Among the tools developed while studying questions of this type are gadget graphs, called

A well-known theorem of Sabidussi shows that a simple -arc-transitive graph can be represented as a coset graph for the group . This pivotal result is the standard way to turn problems about simple arc-transitive graphs into questions about groups. In this paper, the Sabidussi representation is extended to arc-transitive, not necessarily simple graphs which satisfy a local-finiteness condition: namely

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.