Speaker: Jingwei Xu (UIUC)

Title: Injective edge colorings of degenerate graphs and the oriented chromatic number

Abstract: Given a graph G, an injective edge-coloring of G is an edge-labelling φ such that if φ(e) = φ(e'), then no third edge joins an endpoint of e and an endpoint of e'. The injective chromatic index of a graph 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 D, then the injective chromatic index of G is at most O(d^3 log D). Next, we show that if G is a graph of Euler genus g, then the injective chromatic index of G is at most (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(g^6400), improving the previously known upper bound of 2^O(g^(1/2+ ε)) and resolving a conjecture of Aravind and Subramanian.