科研之友
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摘要
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,..., vertical bar E vertical bar 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,..., vertical bar E vertical bar 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.
