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

Abstract

A proper vertex colouring of a graph $G$ is referred to as 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 $\chi_{\rm pcf}(G)$ and $\chi_{\rm pcf}^h(G)$, respectively. It is known that $\chi_{\rm pcf}^h(G)$ can be as large as $(h+1)(\Delta+1)\approx \Delta^2$ for graphs with maximum degree $\Delta$ and $h$ very close to $\Delta$. We provide several new upper bounds for these parameters for graphs with minimum degrees $\delta$ large enough and $h$ detached from $\delta$. In particular we show that $\chi_{\rm pcf}^h(G)\leq (1+o(1))\Delta$ if $\delta\gg\ln\Delta$ and $h\ll \delta$, and that $\chi_{\rm pcf}(G)\leq \Delta+O(\ln \Delta)$ for regular graphs. These specifically refer to the conjecture of Caro, Petru\v{s}evski and \v{S}krekovski that $\chi_{\rm pcf}(G)\leq \Delta+1$ for every connected graph $G$ of maximum degree $\Delta\geq 3$, towards which they proved that $\chi_{\rm pcf}(G)\leq \left\lfloor\frac{5\Delta}{2}\right\rfloor$ if $\Delta\geq 1$.