Brooks-type theorems for relaxations of square colorings

Abstract: The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer $h$, the proper $h$-conflict-free chromatic number of a graph $G$, denoted $\chi_{pcf}^h(G)$, is the minimum $k$ such that $G$ has a proper $k$-coloring where every vertex $v$ has $\min{deg_G(v),h}$ colors appearing exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski put forth a Brooks-type conjecture: if $G$ is a graph with $\Delta(G)\ge 3$, then $\chi_{pcf}^1(G)\leq \Delta(G)+1$. The best known result regarding the conjecture is $\chi_{pcf}^1(G)\leq 2\Delta(G)+1$, which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all $h$, and also enlarge the class of graphs for which the conjecture is known to be true. Our main result is the following: for a graph $G$, if $\Delta(G) \ge h+2$, then $\chi_{pcf}^h(G)\le (h+1)\Delta(G)-1$; this is tight up to the additive term as we explicitly construct infinitely many graphs $G$ with $\chi_{pcf}^h(G)=(h+1)(\Delta(G)-1)$. We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for $h$-dynamic coloring.