The Ramsey number r(G) of a graph G is the smallest integer n such that any 2 colouring of the edges of a clique on n vertices contains a monochromatic copy of G. Determining the Ramsey number of G is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs G for which the value of r(G) is known exactly. One such family consists of large vertex disjoint unions of a fixed graph H, we denote such a graph, consisting of n copies of H by nH. This classical result was proved by Burr, Erd˝os and Spencer in 1975, who showed r(nH) = (2jHj􀀀a(H))n+c, for some c = c(H), provided n is large enough. Since it did not follow from their arguments, Burr, Erd˝os and Spencer further asked to determine the number of copies we need to take in order to see this long term behaviour and the value of c. More than 30 years ago Burr gave a way of determining c(H), which only applies when the number of copies n is triple exponential in jHj. In this paper we give an essentially tight answer to this very old problem of Burr, Erd˝os and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.

A longstanding conjecture of Erd˝os and Simonovits states that for every rational r between 1 and 2 there is a graph H such that the largest number of edges in an H-free graph on n vertices is Q(nr). Answering a question raised by Jiang, Jiang and Ma, we show that the conjecture holds for all rationals of the form 2􀀀a=b with b sufficiently large in terms of a.

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd˝os, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a local girth list assignment: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if G is a planar graph and L is a list assignment for G such that |L(v)| ≥ 3 for all v ∈ V(G); |L(v)| ≥ 4 for every vertex v contained in a 4-cycle; and |L(v)| ≥ 5 for every v contained in a triangle, then G admits an L-colouring.

A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-_free_ if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth. Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $k$-Coloring (for fixed $k$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.

The chromatic number of graphs of maximum degree $\Delta$ has been studied extensively. A greedy coloring procedure shows that any graph has a proper coloring with $\Delta+1$ colors, and this is sharp, as shown by the complete graph of $\Delta+1$ vertices. However, this simple upper bound can often be improved significantly if one excludes some configurations locally or globally. For instance, it is known that triangle-free graphs have much smaller chromatic number, namely $O(\Delta/\log \Delta)$ (Johansson 1996). Consider a vertex $v$, and count the number of edges among its (at most $\Delta$) neighbors. This number is clearly at most ${\Delta \choose 2}$, and it is $0$ if the graph is triangle-free. We say that the graph is $\sigma$-sparse if for each vertex $v$, this number of edges is at most $(1-\sigma){\Delta \choose 2}$. Molloy and Reed proved that $\sigma$-sparse graphs, for $\sigma>0$, can be colored with $(1-\epsilon)\Delta$ colors, for some $\epsilon>0$ depending only on $\sigma$, by a simple randomized procedure: 1. color all the vertices with random colors, 2. uncolor vertices which have received the same color as some of their neighbors, and 3. color the uncolored vertices deterministically (since they have a specific structure, with high probability). This result is a crucial ingredient in a general approach consisting in dividing graphs into $\sigma$-sparse components on one side, and much denser components on the other side, that are colored using different techniques (dense components would typically be colored deterministically, or with random permutations of colors, rather than with random colors). This powerful approach, pioneered by Molloy and Reed, has been rediscovered recently in the field of distributed computing. Moreover, $\sigma$-sparse graphs appear the context of some longstanding open problems in graph coloring. One is a fascinating conjecture of Reed, which asserts that every graph with maximum degree $\Delta$ and clique number $\omega$ admits a proper coloring with at most $\lceil (\Delta+\omega+1)/2 \rceil$ colors. Another is a classic conjecture of Erdős and Nešetřil, which states that the edges of a graph of maximum degree $\Delta$ can be colored with $1.25\Delta^2$ colors in such a way that each color class forms a matching as an induced subgraph. This latter problem can be understood in terms of the chromatic number of squares of line graphs, and such graphs turn out to be $\sigma$-sparse, for a surprising large value of $\sigma$. The authors of the present paper obtain an improved bound on the chromatic number of $\sigma$-sparse graphs, which yields state-of-the-art progress on Reed's conjecture and the Erdős-Nešetřil conjecture. They achieve this through the implementation and analysis of a "random priority assignment" uncoloring procedure in Step 2 above, which allows for a more efficient iteration of Steps 1 and 2 in a semirandom coloring procedure. In this way, they not only improve on previous bounds, but also obtain a bound with a leading term that is asymptotically optimal as $\sigma \to 0$.

The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface. The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result). As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.

Let $p$ and $q$ be positive integers. The $(p, q)$-leaper $L$ is a generalised knight which leaps $p$ units away along one coordinate axis and $q$ units away along the other. Consider a free $L$, meaning that $p + q$ is odd and $p$ and $q$ are relatively prime. We prove that $L$ tours the board of size $4pq \times n$ for all sufficiently large positive integers $n$. Combining this with the recently established conjecture of Willcocks which states that $L$ tours the square board of side $2(p + q)$, we conclude that furthermore $L$ tours all boards both of whose sides are even and sufficiently large. This, in particular, completely resolves the question of the Hamiltonicity of leaper graphs on sufficiently large square boards.