We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let c(ℓ) be the limit of the ratio of the maximum number of cycles of length ℓ in an n-vertex tournament and the expected number of cycles of length ℓ in the random n-vertex tournament, when n tends to infinity. It is well-known that c(3)=1 and c(4)=4/3. We show that c(ℓ)=1 if and only if ℓ

The deck of a graph G is the multiset of cards {G−v:v∈V(G)}. Myrvold (1992) showed that the degree sequence of a graph on n≥7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can be reconstructed from any deck missing O(n/d3) cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar

In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds of finite size. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is a

Thomassen proved that 4-connected planar graphs are Hamilton connected by showing that every 2-connected planar graph G contains a Tutte path P between any two given vertices, that is, every component of G−P has at most three neighbors on P. In this paper, we prove a quantitative version of this result for circuit graphs, a natural class of planar graphs which includes all 3-connected planar graphs

The greedy coloring algorithm shows that a graph of maximum degree at most Δ has chromatic number at most Δ+1, and this is tight for cliques. Much attention has been devoted to improving this “greedy bound” for graphs without large cliques. Brooks famously proved that this bound can be improved by one if Δ≥3 and the graph contains no clique of size Δ+1. Reed's Conjecture states that the “greedy bound”

A quadrangular embedding of a graph in a surface Σ, also known as a quadrangulation of Σ, is a cellular embedding in which every face is bounded by a 4-cycle. A quadrangulation of Σ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in Σ. In this paper we determine n(Σ), the order of a minimal quadrangulation of a surface Σ, for all surfaces, both orientable and nonorientable

One of the major problems in combinatorics is to determine the number of r-uniform hypergraphs (r-graphs) on n vertices which are free of certain forbidden structures. This problem dates back to the work of Erdős, Kleitman and Rothschild, who showed that the number of Kr-free graphs on n vertices is 2ex(n,Kr)+o(n2). Their work was later extended to forbidding graphs as induced subgraphs by Prömel and

Minors play a key role in graph theory, and extremal problems on forbidding minors have attracted appreciable amount of interest in the past decades. In this paper, we focus on spectral extrema of Ks,t-minor free graphs, and determine extremal graphs with maximum spectral radius over all Ks,t-minor free graphs of sufficiently large order. This generalizes and improves several previous results. For

The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every n-vertex graph with at least εn2 edges contains a 1-subdivision of the complete graph on cεn vertices, resolving another old conjecture

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph has treewidth k if it can be decomposed by a sequence of noncrossing cutsets of size at most k into pieces of size at most k+1. The study of hereditary graph classes

A necklace can be considered as a cyclic list of n red and n blue beads in an arbitrary order. In the necklace folding problem the goal is to find a large crossing-free matching of pairs of beads of different colors in such a way that there exists a “folding” of the necklace, that is a partition into two contiguous arcs, which splits the beads of any matching edge into different arcs. We give counterexamples

Given positive integers n⩾2k, the Kneser graph KGn,k is a graph whose vertex set is the collection of all k-element subsets of the set {1,…,n}, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of KGn,k is equal to n−2k+2. In this paper, we study the chromatic number of the random

Let ⊲ be a relation between graphs. We say a graph G is ⊲-ubiquitous if whenever Γ is a graph with nG⊲Γ for all n∈N, then one also has ℵ0G⊲Γ, where αG is the disjoint union of α many copies of G. The Ubiquity Conjecture of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper

We consider locally finite, connected, quasi-transitive graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings' splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. Our result also leads to a characterisation of accessible graphs. We obtain applications

We investigate automorphism groups of planar graphs. The main result is a complete recursive description of all abstract groups that can be realized as automorphism groups of planar graphs. The characterization is formulated in terms of inhomogeneous wreath products. In the proof, we combine techniques from combinatorics, group theory, and geometry. Our result significantly improves the Babai's description

A multigraph drawn in the plane is called non-homotopic if no pair of its edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. Edges are allowed to intersect each other and themselves. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily

In 1973, Erdős and Simonovits asked whether every n-vertex triangle-free graph with minimum degree greater than 1/3⋅n is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for each k, what minimum degree guarantees that a triangle-free graph is k-colourable. This problem has a rich history which culminated in its complete solution by Brandt and Thomassé

Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with n vertices and at most 215n/log2⁡n 3-colorings. We improve his construction, giving examples of such graphs with at most 64nlog9/2⁡3<64n0.731 3-colorings. We conjecture this exponent is optimal.

A conjecture of Erdős from 1967 asserts that any graph on n vertices which does not contain a fixed r-degenerate bipartite graph F has at most Cn2−1/r edges, where C is a constant depending only on F. We show that this bound holds for a large family of r-degenerate bipartite graphs, including all r-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erdős conjecture

Let P(n,m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set [n]:={1,…,n} with m=m(n) edges. We show that in the sparse regime, when m/n≤1, with high probability the maximum degree of P(n,m) takes at most two different values. In contrast, this is not true anymore in the dense regime, when m/n>1, where the maximum degree of P(n,m) is not concentrated on any subset

We show that for every ℓ, there exists dℓ such that every 3-edge-connected graph with minimum degree dℓ can be edge-partitioned into paths of length ℓ (provided that its number of edges is divisible by ℓ). This improves a result asserting that 24-edge-connectivity and high minimum degree provides such a partition. This is best possible as 3-edge-connectivity cannot be replaced by 2-edge connectivity

The hat guessing number is a graph invariant based on a hat guessing game introduced by Winkler. Using a new vertex decomposition argument involving an edge density theorem of Erdős for hypergraphs, we show that the hat guessing number of all outerplanar graphs is less than 2125000. We also define the class of layered planar graphs, which contains outerplanar graphs, and we show that every layered

Recently, Kostochka and Yancey [7] proved that a conjecture of Ore is asymptotically true by showing that every k-critical graph satisfies |E(G)|≥⌈(k2−1k−1)|V(G)|−k(k−3)2(k−1)⌉. They also characterized [8] the class of graphs that attain this bound and showed that it is equivalent to the set of k-Ore graphs. We show that for any k≥33 there exists an ε>0 so that if G is a k-critical graph, then |E(

In 2001, Komlós, Sárközy and Szemerédi proved that, for each α>0, there is some c>0 and n0 such that, if n≥n0, then every n-vertex graph with minimum degree at least (1/2+α)n contains a copy of every n-vertex tree with maximum degree at most cn/log⁡n. We prove the corresponding result for directed graphs. That is, for each α>0, there is some c>0 and n0 such that, if n≥n0, then every n-vertex directed

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing ‘Local Separators’ of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2-separators analogous to the 2-separator theorem; 2. an exact characterisation of graphs

Let G be a simple graph, and let n, Δ(G) and χ′(G) be the order, the maximum degree and the chromatic index of G, respectively. We call G overfull if |E(G)|/⌊n/2⌋>Δ(G), and critical if χ′(H)<χ′(G) for every proper subgraph H of G. Clearly, if G is overfull then χ′(G)=Δ(G)+1 by Vizing's Theorem. The core of G, denoted by GΔ, is the subgraph of G induced by all its maximum degree vertices. Hilton and

The directions of an infinite graph G are a tangle-like description of its ends: they are choice functions that choose a component of G−X for all finite vertex sets X⊆V(G) in a compatible manner. Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably

A subgraph H of an edge-coloured graph is called rainbow if all of the edges of H have different colours. In 1989, Andersen conjectured that every proper edge-colouring of Kn admits a rainbow path of length n−2. We show that almost all optimal edge-colourings of Kn admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture

Let A,B⊆Rd both span Rd such that 〈a,b〉∈{0,1} holds for all a∈A, b∈B. We show that |A|⋅|B|≤(d+1)2d. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane H there is a parallel hyperplane H′ such that H∪H′ contain all vertices. The authors conjectured

We give a more simple proof of the Map Color Theorem for nonorientable surfaces that uses only four constructions of current graphs instead of 12 constructions used in the previous proof. For every i=0,1,2,3, using an index one current graph with cyclic current group, we construct a nonorientable triangular embedding of K12s+3i+1 that can be easily modified into a nonorientable triangular embedding

Let f(d) be the smallest value for which every bridgeless graph G with diameter d admits a strong orientation G⇀ such that the diameter of G⇀ is at most f(d). Chvátal and Thomassen (JCT-B, 1978) established general bounds for f(d) and proved that f(2)=6. Kwok et al. (JCT-B, 2010) showed that 9≤f(3)≤11. In this paper, we determine that f(3)=9.

For r:=(r1,…,rq), an r-factorization of the complete λ-fold h-uniform m-vertex hypergraph λKmh is a partition of the edges of λKmh into F1,…,Fq such that each color class Fi is ri-regular and spanning. We prove two results on embedding factorizations. Previously, these results were only known for a few small values of h, and even then only partially. We show that for n⩾hm and s:=(s1,…,sk), the obvious

Given graphs X and Y with vertex sets V(X) and V(Y) of the same cardinality, the friends-and-strangers graph FS(X,Y) is the graph whose vertex set consists of all bijections σ:V(X)→V(Y), where two bijections σ and σ′ are adjacent if they agree everywhere except for two adjacent vertices a,b∈V(X) such that σ(a) and σ(b) are adjacent in Y. The most fundamental question that one can ask about these f

For positive integers d0 and any “not too small” p, we prove that a random k-uniform hypergraph G with n vertices and edge probability p typically has the property that every spanning subgraph of G with minimum d-degree at least (1+ε)md(k,n)p has a perfect matching. One interesting aspect of our proof is a “non-constructive” application of the absorbing method, which allows us to prove a bound in terms

Let G be a graph with chromatic number χ, maximum degree Δ and clique number ω. Reed's conjecture states that χ≤⌈(1−ε)(Δ+1)+εω⌉ for all ε≤1/2. It was shown by King and Reed that, provided Δ is large enough, the conjecture holds for ε≤1/130,000. In this article, we show that the same statement holds for ε≤1/26, thus making a significant step towards Reed's conjecture. We derive this result from a general

Hakimi, Schmeichel, and Thomassen (1979) [10] conjectured that every 4-connected planar triangulation G on n vertices has at least 2(n−2)(n−4) Hamiltonian cycles, with equality if and only if G is a double wheel. In this paper, we show that every 4-connected planar triangulation on n vertices has Ω(n2) Hamiltonian cycles. Moreover, we show that if G is a 4-connected planar triangulation on n vertices

Over recent years there has been much interest in both Turán and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect H-tiling in an ordered graph. In the (unordered) graph

Let M be a 3-connected matroid and let F be a field. Let A be a matrix over F representing M and let (G,B) be a biased graph representing M. We characterize the relationship between A and (G,B), settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation (G,B) of M, A is projectively equivalent to a canonical matrix representation arising

We prove that there exists C>0 such that any (n+Ck)-vertex tournament contains a copy of every n-vertex oriented tree with k leaves, improving the previously best known bound of n+O(k2) vertices to give a result tight up to the value of C. Furthermore, we show that, for each k, there exists n0, such that, whenever n⩾n0, any (n+k−2)-vertex tournament contains a copy of every n-vertex oriented tree with

The characterization of rigid graphs in Rd for d≥3 is a major open problem in rigidity theory. The same holds for globally rigid graphs. In this paper our goal is to give necessary and/or sufficient conditions for the (global) rigidity of the square G2 (and more generally, the power Gk) of a graph G in Rd, for some values of k,d. Our work is motivated by some results and conjectures of M. Cheung and

A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two distinct triangles. These were studied at length by Braß [6] (2004) and by Aronov, Dujmović, Morin, Ooms, and da Silveira [2] (2019). We determine the extremal functions exactly for seven of the eight configurations. The

We introduce the following combinatorial problem. Let G be a triangle-free regular graph with edge density ρ. (In this paper all densities are normalized by n,n22 etc. rather than by n−1,(n2),…) What is the minimum value a(ρ) for which there always exist two non-adjacent vertices such that the density of their common neighbourhood is ≤a(ρ)? We prove a variety of upper bounds on the function a(ρ) that

Hadwiger's conjecture states that every Kt-minor free graph is (t−1)-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd Kt-minor is (t−1)-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form

Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from

Let k and d be positive integers such that k≥d+2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper colouring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist

We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the r-colour spanning-tree discrepancy of a graph G is equal, up to a constant, to the minimum s such that G can be separated into r equal parts by deleting s vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy

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. There is an obvious necessary degree condition for the immersion containment: if G contains H as an immersion, then for every integer k, the number of vertices of degree at least k in G is at least the number of vertices of degree at least k in H. In this

We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 “templates” (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only if it contains a subgraph matching one of the templates. As a consequence, we show every triangle-free graph drawn in the torus with edge-width

Huang proved that every set of more than half the vertices of the d-dimensional hypercube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs

It is proven that for any integer g≥0 and k∈{0,…,10}, there exist infinitely many 5-regular graphs of genus g containing a 1-factorisation with exactly k pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For g=0 and k=10, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's “marriage” operation, we discuss two gluing techniques aimed at producing graphs of

The celebrated Erdős–Stone–Simonovits theorem characterizes the asymptotic maximum edge density in F-free graphs as 1−1/(χ(F)−1)+o(1), where χ(F) is the minimum chromatic number of a graph in F. In [6, Examples 25 and 31], it was shown that this result can be extended to the general setting of graphs with extra structure: the asymptotic maximum edge density of a graph with extra structure without some

One of the most fundamental results in structural graph theory is the “two-paths theorem” that characterizes 2-linkage by planarity. As an extension of the theorem, we consider the following problem for a fixed graph H with four vertices: Given a graph G and an injective map from V(H) to V(G), is there a subdivision of H in G with four branch vertices specified by the map? Hence the case H=2K2 corresponds

Given a finite relational language L, a hereditary L-property is a class H of finite L-structures closed under isomorphism and substructure. The speed of H is the function which sends an integer n≥1 to the number of distinct elements in H with underlying set {1,...,n}. In this paper we give a description of many new jumps in the possible speeds of a hereditary L-property, where L is any finite relational

The canonical double cover B(X) of a graph X is the direct product of X and K2. If Aut(B(X))≅Aut(X)×Z2 then X is called stable; otherwise X is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. A circulant is a Cayley graph on a cyclic group. Qin et al. (2019) [18] conjectured that there are no nontrivially

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture. Let C be a clean tangled clutter. It was recently proved that C has a fractional packing

An (n,s,q)-graph is an n-vertex multigraph in which every s-set of vertices spans at most q edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in (n,s,q)-graphs. We give a general lower

In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A q-matroid is a q-analogue of a matroid where subspaces play the role of the subsets in the classical theory. In this article we establish cryptomorphisms of q-matroids. In doing so we highlight the difference between classical theory and

Let Hr(n,p) denote the maximum number of Hamiltonian cycles in an n-vertex r-graph with density p∈(0,1). The expected number of Hamiltonian cycles in the random r-graph model Gr(n,p) is E(n,p)=pn(n−1)!/2 and in the random graph model Gr(n,m) with m=p(nr) it is, in fact, slightly smaller than E(n,p). For graphs, H2(n,p) is proved to be only larger than E(n,p) by a polynomial factor and it is an open

For any even integer k≥6, integer d such that k/2≤d≤k−1, and sufficiently large n∈(k/2)N, we find a tight minimum d-degree condition that guarantees the existence of a Hamilton (k/2)-cycle in every k-uniform hypergraph on n vertices. When n∈kN, the degree condition coincides with the one for the existence of perfect matchings provided by Rödl, Ruciński and Szemerédi (for d=k−1) and Treglown and Zhao

A signed bipartite (simple) graph (G,σ) is said to be C−4-critical if it admits no homomorphism to C−4 (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study of C−4-critical signed graphs, we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to C−4. In particular, the 4-color theorem