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 $\chi_\beta(G) = \mathcal{O}(\Delta(G)^{1 + 1/\beta})$, 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 $\chi_\beta(G) = \mathcal{O}(\Delta(G)^{1 + 1/\beta} / (\log\Delta(G))^{1/\beta})$. 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 $\Delta$ and girth such that $\chi_\beta(G) = \Omega(\Delta^{1 + 1/\beta} / (\log\Delta)^{1/\beta})$. The upper bounds are obtained via a sparse hypergraph colouring theorem of Li and Postle.