[r,s,t]-Coloring of Trees and Bipartite Graphs
Résumé
Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. Given non negative integers $r$, $s$ and $t$, an [$r,s,t$]-\emph{coloring} of a graph $G$ is a function $c$ from $V(G) \cup E(G)$ to the color set $\{ 0, 1,..., k-1 \}$ such that $|c(v_i) - c(v_j)|\geq r$ for every two adjacent vertices $v_i$, $v_j$ $\in V$, $|c(e_i) - c(e_j)|\geq s$ for every two adjacent edges $e_i$, $e_j$ $\in E$, and $|c(v_i) - c(e_j)|\geq t$ for every vertex $v_i$ and its incident edge $e_j$. Thus, an [$r,s,t$]-coloring is a generalization of the total coloring and the classical vertex and edge colorings of graphs. The [$r,s,t$]-coloring can have many applications in different fields like scheduling \cite{Kemnitz1}, channel assignment problem \cite{Bazzaro},.... The [$r,s,t$]\textit{-chromatic number} $\chi_{r,s,t}(G)$ of $G$ is the minimum $k$ such that $G$ admits an [$r,s,t$]-coloring. In our paper, we give exact values (or bounds in one case) of the [$r,s,t$]-chromatic number of stars, for every positive $r$, $s$ and $t$. We also provide exact values and some tight bounds of this parameter for trees and bipartite graphs.