Parity Edge-Coloring of Graphs
Abstract
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'(K_n). Specifically, if m is the least power of two that is as large as n, then p'(K_n) 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 K_{2,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.
Cite
@article{arxiv.math/0602341,
title = {Parity Edge-Coloring of Graphs},
author = {David P. Bunde and Kevin Milans and Douglas B. West and Hehui Wu},
journal= {arXiv preprint arXiv:math/0602341},
year = {2007}
}
Comments
23 pages, 0 figures