Weighted antimagic labeling

A graph G=(V,E) is weighted- k-antimagic if for each w:V→R, there is an injective function f:E→{1,…,|E|+k} such that the following sums are all distinct: for each vertex u, ∑_v:uv∈Ef(uv)+w(u). When such a function f exists, it is called a (w,k)-antimagic labeling of G. A connected graph G is antimagic if it has a (w₀,0)-antimagic labeling, for w₀(u)=0, for each u∈V. In this work, we prove that all the complete bipartite graphs K_p,q, are weighted-0-antimagic when 2≤p≤q and q≥3. Moreover, an algorithm is proposed that computes in polynomial time a (w,0)-antimagic labeling of the graph. Our result implies that if H is a complete partite graph, with H≠K_1,q, K_2,2, then any connected graph G containing H as a spanning subgraph is antimagic.