On conflict-free proper colourings of graphs without small degree vertices

Abstract

A proper vertex colouring of a graph G is conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called h-conflict-free if there are at least h such colours for each vertex of G. The least numbers of colours in such colourings of G are denoted by χpcf(G) and χpcfh(G), respectively. The latter parameter may be regarded as a natural relaxation of the 2-chromatic number, χ2(G), i.e. the least number of colours in a proper colouring of the square of a given graph G. It is known that χpcfh(G) can be as large as (h+1)(Δ+1)≈Δ2 for graphs with maximum degree Δ and h very close to Δ. We provide several new upper bounds for these parameters for graphs with minimum degrees δ large enough and h of smaller order than δ. In particular, we show that χpcfh(G)⩽(1+o(1))Δ if δ≫ln⁡Δ and h≪δ, and that χpcf(G)⩽Δ+O(ln⁡Δ) for regular graphs. These are related with the conjecture of Caro, Petruševski and Škrekovski that χpcf(G)⩽Δ+1 for every connected graph G of maximum degree Δ⩾3, towards which they proved that χpcf(G)⩽⌊5Δ2⌋ if Δ⩾1.