Distance edge coloring by total labeling of graphs

Distance edge-coloring total labeling of a connected graph G is an assignment f of non negative integers to the vertices and edges of G such that w(e) not equal w(e') if d(e, e') <= l for any two edges e and e' of G, where w(e) denotes the weight of an edge e = uv and is defined by: w(uv) = f (u) + f (v) + f (uv) and d(e, e') is the distance between e an e' in G. In this paper, we propose a lower and an upper bounds for the chromatic number of the distance edge coloring by total labeling of general graphs for a positive integer 0 <= l <= diam(G)-1. Moreover, we prove exact values for this parameter in the case of paths, cycles and spiders.