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The odd chromatic index of almost all graphs

An odd coloring of a graph G is an assignment of colors to the edges of G in a way that each vertex is incident to either zero or an odd number of edges of each color. The minimum number of colors needed in such a coloring is called the odd chromatic index of G, and it is denoted by χo(G). This notion was introduced by Pyber, who showed that χo(G) ≤ 4 holds for every graph G. In this paper, we show that almost every graph on an even (resp. odd) number of vertices satisfies χo(G) = 2 (resp. χ ′ o(G) = 3).