Graphs admitting antimagic labeling for arbitrary sets of positive numbers

Abstract

Hartsfield and Ringel in 1990 conjectured that any connected graph with q≥2 edges has an edge labeling f with labels in the set {1,…,q}, such that for every two distinct vertices u and v, fu≠fv, where fv=∑e∈E(v)f(e), and E(v) is the set of edges of the graph incident to vertex v.We say that a graph G=(V,E), with q edges, is universal antimagic, if for every set B of q positive numbers there is a bijection f:E→B such that fu≠fv, for any two distinct vertices u and v. It is weighted universal antimagic if for any vertex weight function w and every set B of q positive numbers there is a bijection f:E→B such that w(u)+fu≠w(v)+fv, for any two distinct vertices u and v.In this work we prove that paths, cycles, and graphs whose connected components are cycles or paths of odd lengths are universal antimagic. We also prove that a split graph and any graph containing a complete bipartite graph as a spanning subgraph is universal antimagic. Surprisingly, we are also able to prove that any graph containing a complete bipartite graph Kn,m with n,m≥3 as a spanning subgraph is weighted universal antimagic. From all the results we can derive effective methods to construct the labelings.