Interval cyclic edge-colorings of graphs

Abstract

A proper edge-coloring α of a graph G with colors 1,…,t is called an interval cyclict-coloring if all colors are used, and the colors of edges incident to each vertex v of G either form an interval of integers or the set {1,…,t}∖{α(e):eisincidenttov} is an interval of integers. A graph G is interval cyclically colorable if it has an interval cyclic t-coloring for some positive integer t. The set of all interval cyclically colorable graphs is denoted by Nc. For a graph G∈Nc, the least and the greatest values of t for which it has an interval cyclic t-coloring are denoted by wc(G) and Wc(G), respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if G is a triangle-free graph with at least two vertices and G∈Nc, then Wc(G)≤|V(G)|+Δ(G)−2. We also obtain some bounds on wc(G) and Wc(G) for various classes of graphs. Finally, we give two methods for constructing of interval cyclically non-colorable graphs.