Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non-negative integer. We say G is k-antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted-k-antimagic if for any vertex weight function w: V→ℕ, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well-known conjecture asserts that every connected graph G≠K2 is 0-antimagic. On the other hand, there are connected graphs G≠K2 which are not weighted-1-antimagic. It is unknown whether every connected graph G≠K2 is weighted-2-antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted-2-antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic. We also prove that every connected graph G≠K2 on n vertices is weighted- ⌊3n/2⌋-antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.