On (0,1)-relaxed Strong Edge-coloring Of Sparse Graphs

Let G=(V(G),E(G))be a graph.Ak-edge-coloring of a graph G is a mapping c:E(G)?[k]([k]is a set of colors).For two nonnegative integers s and t,an(s,t)-relaxed strong k-edge-coloring is a k-edge-coloring 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?'(s,t)(G),is the minimum number k for which G has an(s,t)-relaxed strong k-edge-coloring.In this paper,we prove that if G is a graph with mad(G)<3,and ? ? 7,then?'(0,1)(G)? 3?-1.In addition,we prove that if G is a planar graph with ?? 4 and g? 7,then ?'(0.1)(G)?[5?/2].