MAD Phase Transitions in the Oriented Chromatic Number

For an oriented graph $G$ the oriented chromatic number of $G$, written $\chi_o(G)$, is the least integer $t$ such that $G$ has a homomorphism to a tournament on $t$ vertices. The oriented chromatic number of a simple graph $H$ is the maximum oriented chromatic number over all orientations of $H$. Borodin, Kostochka, Ne{\v{s}}et{\v{r}}il, Raspaud, and Sopena proved in 1999 that for all $\epsilon>0$, graphs with maximum average degree less than $4-\epsilon$ have bounded oriented chromatic number. This is in some sense optimal, because $1$-subdivisions of cliques demonstrate that there exists graphs with maximum average degree strictly less than $4$ and oriented chromatic number $\Omega(\sqrt{n})$. We prove that for every positive integer $d$, every $d$-degenerate graph $G$ with sufficiently large order satisfies $\chi_o(G) \leq 6(1+\frac{9d^2}{4})^{\frac{1}{2}}d 8^{d}\sqrt{n}$. This implies for a fixed $r\geq 4$, the optimal bound for the oriented chromatic number of graphs with maximum average degree less than $r$ is $\Theta(\sqrt{n})$. This complements a bound of Wood, who showed that for all $n$ vertex graphs $\chi_o \leq 2\Delta\sqrt{n-1}$ where $\Delta$ is the maximum degree.