Colouring graphs when the number of colours is nearly the maximum degree

Abstract

We consider for graphs of maximum degree Δ, the problem of determining whether kG) > Δ-k for various values of k. We obtain sharp theorems characterizing when the barrier to Δ-k colourability must be a local condition, i.e. a small subgraph, and when it can be global. We also show that for large fixed Δ, this problem is either NP-complete or can be solved in linear time, and we determine precisely which values of k correspond to each case prove that Hitting Set with sets of size B is hard to approximate to within a factor $B^{1/19}$. The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B=2. The relationship between hardness of approximation and set size seems to have not been explored before.