On (s, t)-relaxed strong edge-coloring of graphs

Abstract

In this paper, we introduce a new relaxation of strong edge-coloring. Let G be a graph. For two nonnegative integers s and t, an (s, t)-relaxed strong k-edge-coloring is an assignment of k colors to the edges of G, such that for any edge e, there are at most s edges adjacent to e and t edges which are distance two apart from e assigned the same color as e. The (s, t)-relaxed strong chromatic index, denoted by $${\chi '}_{(s,t)}(G)$$źź(s,t)(G), is the minimum number k of an (s, t)-relaxed strong k-edge-coloring admitted by G. This paper studies the (s, t)-relaxed strong edge-coloring of graphs, especially trees. For a tree T, the tight upper bounds for $${\chi '}_{(s,0)}(T)$$źź(s,0)(T) and $${\chi '}_{(0,t)}(T)$$źź(0,t)(T) are given. And the (1, 1)-relaxed strong chromatic index of an infinite regular tree is determined. Further results on $${\chi '}_{(1,0)}(T)$$źź(1,0)(T) are also presented.