The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we announceâ€¦ (More)

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. Inâ€¦ (More)

This is a survey of Hadwigerâ€™s conjecture from 1943, that for all t â‰¥ 0, every graph either can be t-coloured, or has a subgraph that can be contracted to the com plete graph ont +1 vertices. This isâ€¦ (More)

Given k pairs of vertices (si, ti) (1 â‰¤ i â‰¤ k) of a digraph G, how can we test whether there exist vertex-disjoint directed paths from si to ti for 1 â‰¤ i â‰¤ k? This is NP-complete in general digraphs,â€¦ (More)

A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for all Î½ > 0, every triangle-free graph with sufficiently large chromatic number contains holes of Î½â€¦ (More)

An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. GyÃ¡rfÃ¡s made the conjecture that for all t there exists n such that every graph with no Ktâ€¦ (More)

GyÃ¡rfÃ¡s conjectured in 1985 that for all k, l, every graph with no clique of size more than k and no odd hole of length more than l has chromatic number bounded by a function of k, l. We prove threeâ€¦ (More)

Given two graphs, G and H, we say that H is an induced subgraph of G if V (H) âŠ† V (G), and two vertices of H are adjacent if and only if they are adjacent in G. Let F be a (possibly infinite) familyâ€¦ (More)

In 1961, Claude Berge proposed the \strong perfect graph conjecture", probably the most beautiful open question in graph theory. It was answered just before his death in 2002. This is an overview ofâ€¦ (More)

Wound infection after carotid endarterectomy is an extremely uncommon complication. This complication occurred in two of our patients in a series of 170 consecutive carotid endarterectomies over theâ€¦ (More)