A Note on K-Δ+1-Free Precolouring with Δ Colours

Let G be a simple graph of maximum degree ∆ > 3, not containing K∆+1, and L a list assignment to V (G) such that |L(v)| = ∆ for all v ∈ V (G). Given a set P ⊂ V (G) of pairwise distance at least d then Albertson, Kostochka and West (2004) and Axenovich (2003) have shown that every L-precolouring of P extends to a L-colouring of G provided d > 8. Let K− ∆+1 denote the graph K∆+1 with one edge removed. In this paper, we consider the problem above and the effect on the pairwise distance required when we additionally forbid either K− ∆+1 or K∆ as a subgraph of G. We have the corollary that an extra assumption of 3-edge-connectivity in the above result is sufficient to reduce this distance from 8 to 4. This bound is sharp with respect to both the connectivity and distance. In particular, this corrects the results of Voigt (2007, 2008) for which counterexamples are given.