Distance-edge-coloring of power graphs
Résumé
The \ell-distance-edge-coloring is a generalization of the edge-coloring that tries to assign a color from 1 to k to each edge such that any two edges of distance at most \ell have distinct colors. The minimum number of colors used to color a graph with an \ell-distance-edge-coloring is called the \ell-chromatic index. Solving the \ell-distance-edge-coloring problem is different from determining the chromatic index, by Vizing's theorem, for its power graph. The \ell-distance-edge-coloring is thus an original problem which is NP-hard in general. We study the \ell-distance-edge-coloring problem in some classes of graphs in order to bring a new insight on the relative difficulties of this problem. We focus on the \ell-chromatic index for some classes of p-th power graphs. We present, for any integer \ell\geq0, the exact values for the \ell-chromatic index for power graphs of paths and complete k-ary trees, and bounds and exact values for the \ell-chromatic index for power graphs of cycles and general trees. Furthermore, we propose a polynomial-time coloring algorithm, for each studied class of graph, that satisfies the \ell-distance-edge-coloring.