Forbidden Subgraph Colorings and the Oriented Chromatic Number

Abstract

We present an improved upper bound of $O(d^{1+\frac{1}{m-1}})$ for the $(2,\mathcal{F})$-subgraph chromatic number $\chi_{2,\mathcal{F}}(G)$ of any graph G of maximum degree d. Here, m denotes the minimum number of edges in any member of $\mathcal{F}$. This bound is tight up to a (logd)^{1/(m − 1)} multiplicative factor and improves the previous bound presented in [1].

We also obtain a relationship connecting the oriented chromatic number χ _o (G) of graphs and the $(j,\mathcal{F})$-subgraph chromatic numbers $\chi_{j,\mathcal{F}}(G)$ introduced and studied in [1]. In particular, we relate oriented chromatic number and the (2,r)-treewidth chromatic number and show that $\chi_o(G) \leq k\left((r+1)2^r\right)^{k-1}$ for any graph G having (2,r)-treewidth chromatic number at most k. The latter parameter is the least number of colors in any proper vertex coloring which is such that the subgraph induced by the union of any two color classes has treewidth at most r.

We also generalize a result of Alon, et. al. [2] on acyclic chromatic number of graphs on surfaces to $(2,\mathcal{F})$-subgraph chromatic numbers and prove that $\chi_{2,\mathcal{F}}(G) = O(\gamma^{m/(2m-1)})$ for some constant m depending only on $\mathcal{F}$. We also show that this bound is nearly tight. We then use this result to show that graphs of genus g have oriented chromatic number at most $2^{O(g^{1/2 + \epsilon})}$ for every fixed ε> 0. This improves the previously known bound of $2^{O(g^{4/7})}$. We also refine the proof of a bound on χ _o (G) obtained by Kostochka, et. al. in [3] to obtain an improved bound on χ _o (G).