Injective edge colorings of degenerate graphs and the oriented chromatic number

Abstract

Given a graph $G$, an injective edge-coloring of $G$ is a function $\psi :E\left(G\right)\to N$ such that if $\psi \left(e\right)=\psi \left(e^{\prime }\right)$, then no third edge joins an endpoint of $e$ and an endpoint of $e^{\prime }$. The injective chromatic index of a graph $G$, written ${\chi }_{inj}^{\prime }\left(G\right)$, is the minimum number of colors needed for an injective edge coloring of $G$. In this paper, we investigate the injective chromatic index of certain classes of degenerate graphs. First, we show that if $G$ is a $d$-degenerate graph of maximum degree $\Delta$, then ${\chi }_{inj}^{\prime }\left(G\right)=O(d^{3}log\Delta )$. Next, we show that if $G$ is a graph of Euler genus $g$, then ${\chi }_{inj}^{\prime }\left(G\right)\le (3+o\left(1\right))g$, which is tight when $G$ is a clique. Finally, we show that the oriented chromatic number of a graph is at most exponential in its injective chromatic index. Using this fact, we prove that the oriented chromatic number of a graph embedded on a surface of Euler genus $g$ has oriented chromatic number at most $O\left(g^{6400}\right)$, improving the previously known upper bound of $2^{O\left(g^{\frac{1}{2}+\varepsilon }\right)}$ and resolving a conjecture of Aravind and Subramanian.