Distance edge coloring by total labelings of graphs

Abstract

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 ω(e) ≠ ω(e') if d(e, e') ≤ ℓ for any two edges e and e' of G, where ω(e) denotes the weight of an edge e = uv and is defined by: ω(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.