Injective edge colorings of degenerate graphs and the oriented chromatic number

Given a graph G, an injective edge-coloring of G is a function fr : E(G) - N such that if fr(e) = fr(e '), then no third edge joins an endpoint of e and an endpoint of e '. The injective chromatic index of a graph G, written chi ' inj(G), 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 triangle, then chi ' inj(G) = O(d3 log triangle). Next, we show that if G is a graph of Euler genus g, then chi ' inj(G) <= (3+ o(1))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(g6400), improving the previously known upper bound of 2O(g2 +epsilon) and resolving a conjecture of Aravind and Subramanian. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).