List Colouring Constants of Triangle Free Graphs

In this paper we prove a result about vertex list colourings which in particular shows that a conjecture of the second author (1999, Journal of Graph Theory 31, 149-153) is true for triangle free graphs of large maximum degree. There exists a constant K such that the following holds: Given a graph G and a list assignment L to vertices of G, assigning a list of available colours L ( v ) to each vertex v ∈ V ( G ) , such that | L ( v ) | = K Δ log ( Δ ) , then there exists a proper list colouring of vertices of G provided that for each colour c, the graph induced by all vertices v with c ∈ L ( v ) is triangle free and has maximum degree at most Δ.