Balanced edge colorings

This paper contains two principal results. The first proves that any graph G can be given a balanced proper edge coloring by kappa colors for any kappa > x' (G). Here balanced means that the number of vertices incident with any set of d colors is essentially fixed for each d, that is, for two different d-sets of colors the number of vertices incident with each of them can differ by at most 2. The second result gives upper bounds for the vertex-distinguishing edge chromatic number of graphs G with few vertices of low degree. In particular, it proves a conjecture of Burris and Schelp in the case when Delta(G) greater than or equal to root2\V(G)\ + 4 and delta(G) greater than or equal to 5. (C) 2003 Elsevier Inc. All rights reserved.