Distance edge coloring by total labeling of graphs
Résumé
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 $w(e) \neq w(e') \text{ if } d(e,e') \leq \ell$
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 \leq \ell \leq \text{diam}(G) - 1$. Moreover, we prove exact values for this parameter in the case of paths, cycles and spiders.