Vertex-coloring edge-weighting of complete r-partite graphs - Li, Yunyun; Xu, Changqing (Utilitas Mathematica, 2013)
A k-edge-weighting of graph G is a mapping w : E(G) → {1, 2, ···, k}. It is called a vertex-coloring k-edge-weighting if the weighted degrees dw(v) = ΣuεN(v) w(uv) of the vertices yield a proper vertex coloring of graph G. Let μ(G) be the minimum k for which G has a vertex-coloring k-edge-weighting. In this note we prove that μ(Kn1, n2, ···, nr) is 1 if n1, n2, ···, nr are pairwise distinct, is 3 if n1 = n2 = · · · = nr = 1, and is 2 otherwise.