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 k colors for any k⩾χ′(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 Δ(G)⩾2|V(G)|+4 and δ(G)⩾5.