Hadwiger's conjecture asserts that if a simple graph G has no K t+1 minor, then its vertex set V (G) can be partitioned into t stable sets. This is still open, but we prove under the same hypotheses that V (G) can be partitioned into t sets X 1 , . . . , X t , such that for 1 ≤ i ≤ t, the subgraph induced on X i has maximum degree at most a function of t. This is sharp, in that the conclusion becomes false if we ask for a partition into t − 1 sets with the same property.

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree ∆. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.

It was conjectured by the third author in about 1973 that every d-regular planar graph (possibly with parallel edges) can be d-edge-coloured, provided that for every odd set X of vertices, there are at least d edges between X and its complement. For d = 3 this is the four-colour theorem, and the conjecture has been proved for all d ≤ 7, by various authors. Here we prove it for d = 8.

We provide a quasilinear time algorithm for the p-center problem with an additive error less than or equal to 3 times the input graph's hyperbolic constant. Specifically, for the graph G = (V, E) with n vertices, m edges and hyperbolic constant δ, we construct an algorithm for p-centers in time O(p(δ + 1)(n + m) log(n)) with radius not exceeding r p + δ when p ≤ 2 and r p + 3δ when p ≥ 3, where r p are the optimal radii. Prior work identified p-centers with accuracy r p + δ but with time complexity O((n 3 log n + n 2 m) log(diam(G))) which is impractical for large graphs.

No abstract