Although the theory of random graphs is one of the youngest branches of graph theory, in
importance it is second to none. It began with some sporadic papers of Erdős in the 1940s …

Over the past decade, many major advances have been made in the field of graph colouring
via the probabilistic method. This monograph provides an accessible and unified treatment …

Suppose t1,..., tn are independent random variables which take value either 0 or 1, and Y is
a multi-variable polynomial in ti's with positive coefficients. We give a condition which …

We prove that every triangle-free graph with maximum degree Δ has list chromatic number
at most (1+ o (1)) Δ ln⁡ Δ. This matches the best-known upper bound for graphs of girth at …

There is a great variety of colouring concepts and results in the literature. Here our focus is
to survey results on vertex colourings of graphs defined in terms of forbidden induced …

Extremal problems concerned with hypergraph colouring first arose in connection with
classical investigations in the 1920-30s which gave rise to Ramsey theory. Since then, this …

We develop an improved bound for the chromatic number of graphs of maximum degree Δ
under the assumption that the number of edges spanning any neighbourhood is at most for …

Abstract The Erdős--Faber--Lovász conjecture (posed in 1972) states that the chromatic
index of any linear hypergraph on n vertices is at most n. In this paper, we prove this …

It is shown that the chromatic number of any graph with maximum degree d in which the
number of edges in the induced subgraph on the set of all neighbors of any vertex does not …

A remarkable connection has been established for antiferromagnetic 2-spin systems,
including the Ising and hard-core models, showing that the computational complexity of …