Parity Edge-Coloring of Graphs

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In a graph whose edges are colored, a parity walk is a walk that uses each color an even number of times. The parity edge chromatic number p(G) of a graph G is the least k so that there is a coloring of E(G) using k colors that does not contain a parity path. The strong parity edge chromatic number p'(G) of G is the least k so that there is a coloring of E(G) using k colors with the property that every parity walk is closed. Our main result is to determine p'(Kₙ). Specifically, if m is the least power of two that is as large as n, then p'(Kₙ) has value m - 1. As a corollary, we strengthen a special case of an old result of Daykin and Lovasz. Other results include determining p(G) and p'(G) whenever G is a path, cycle, or of the form K₂,ₙ, and an upper bound on p'(G) for the case that G is a complete bipartite graph. We conclude with a sample of open problems.