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 ≠ K 2 is 0‐antimagic. On the other hand, there are connected graphs G ≠ K 2 which are not weighted‐1‐antimagic. It is unknown whether every connected graph G ≠ K 2 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 ≠ K 2 on n vertices is weighted‐ ⌊3 n /2⌋‐antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory