Edge-coloring vertex-weighting of graphs

Abstract

Let G = (V (G),E(G)) be a simple, finite and undirected graph of order n. A k-vertex weighting of a graph G is a mapping w: V (G) → {1,…, k}. A k-vertex weighting induces an edge labeling fw: E(G) → N such that fw(uv) = w(u) + w(v). Such a labeling is called an edge-coloring k-vertex weighting if fw(e)≠ fw(e′) for any two adjacent edges e and e′. Denote by μ′(G) the minimum k for G to admit an edge-coloring k-vertex weighting. In this paper, we determine μ′(G) for some classes of graphs.