On Restricted Colorings of text{(d,s)}-Edge Colorable Graphs

Abstract

A cycle is 2-colored if its edges are properly colored by two distinct colors. A (d, s)-edge colorable graphG is a d-regular graph that admits a proper d-edge coloring in which every edge of G is in at least s-1 2-colored 4-cycles. Given a (d, s)-edge colorable graph G and a list assigment L of forbidden colors for the edges of G satisfying certain sparsity conditions, we prove that there is a proper d-edge coloring of G that avoids L, that is, a proper edge coloring varphi of G such that varphi (e) notin L(e) for every edge e of G. Additionally, this paper also contains a discussion of graphs belonging to the family of (d, s)-edge colorable graphs.