Local edge coloring of graphs

Let G = (V, E) be a graph. A local edge coloring of G is a proper edge coloring c:E -> N such that for each subset S of E(G) with 2 <= vertical bar S vertical bar <= 3, there exist edges e,f is an element of S such that vertical bar c(e)-c(f)vertical bar >= n(s), where n(s) is the number of copies of P-3 in the edge induced subgraph S. The maximum color assigned by a local edge coloring c to an edge of G is called the value of c and is denoted by chi(l)'(c). The local edge chromatic number of G is chi(l)'(G)=min{chi(l)'(c)}, where the minimum is taken over all local edge colorings c of G. In this article, we derive bounds and many results based on local edge coloring.