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, f(u) not equal= f(v), where f(v) = Sigma(e is an element of 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 f(u) not equal= f(v), 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) + f(u) not equal= w(v) + f(v), 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 K-n,K-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.