Vertex-coloring edge-weightings of graphs

A k-edge-weighting of a graph G is a mapping w: E(G) → {1, 2, . . ., k}. An edge-weightingw induces a vertex coloring f_w: V (G)→ N defined by f_w(v) = ∑_v∈e w(e). An edge-weighting w is vertex-coloring if f_w(u) ≠ f_w(v) for any edge uv. The current paper studies the parameter μ(G), which is the minimum k for which G has a vertex-coloring k-edgeweighting. Exact values of μ(G) are determined for several classes of graphs, including trees and r-regular bipartite graph with r ≥ 3.