r-Strong edge colorings of graphs

If c : E → {1, 2, . . . , k} is a proper edge coloring of a graph G = (V,E) then the palette S(v) of a vertex v ∈ V is the set of colors of the incident edges: S(v) = {c(e) : e = vw ∈ E}. An edge coloring c distinguishes vertices u and v if S(u) 6= S(v). A d-strong edge coloring of G is a proper edge coloring that distinguishes all pairs of vertices u and v with distance d(u, v) ≤ d. The minimum number of colors of a d-strong edge coloring is called d-strong chromatic index χ d (G) of G. We will present some general results on d-strong edge colorings as well as specific results for paths and cycles and a general conjecture.