Frugal colourings of graphs via sparse hypergraph colouring

A proper colouring of a graph $G$ is $\beta$-frugal if every colour appears at most $\beta$ times in the neighbourhood of each vertex. Let ${\chi }_{\beta }(G)$ denote the minimum number of colours needed for a $\beta$-frugal colouring of $G$. For a fixed value of $\beta$, Hind et al. showed that ${\chi }_{\beta }(G)=\mathcal{O}(\mathrm{\Delta }(G{)}^{1+1\mathrm{/}\beta })$, and a construction of Alon certifies the tightness of this upper bound up to a constant factor. We show that, for all fixed $\beta \ge 2$ and $t\ge 2$, if $G$ does not contain $C_{2t}$ as a subgraph, or if $G$ does not contain $K_{\beta ,t}$ as a subgraph, then ${\chi }_{\beta }(G)=\mathcal{O}(\mathrm{\Delta }(G{)}^{1+1\mathrm{/}\beta }\mathrm{/}(\log \mathrm{\Delta }(G){)}^{1\mathrm{/}\beta })$. Furthermore, we show that these upper bounds are tight up a constant factor due to the existence of graphs $G$ with arbitrarily large maximum degree $\mathrm{\Delta }$ and girth such that ${\chi }_{\beta }(G)=\mathrm{\Omega }({\mathrm{\Delta }}^{1+1\mathrm{/}\beta }\mathrm{/}(\log \mathrm{\Delta }{)}^{1\mathrm{/}\beta })$. The upper bounds are obtained via a sparse hypergraph colouring theorem of Li and Postle.