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Antimagic labelling of vertex weighted graphs. (English) Zbl 1244.05192
Summary: Suppose \(G\) is a graph, \(k\) is a non-negative integer. We say \(G\) is \(k\)-antimagic if there is an injection \(f: E\to \{1, 2, \dots , |E| + k\}\) such that for any two distinct vertices \(u\) and \(v\), \(\sum_{e\in E(v)}f(e)\neq\sum_{e\in E(u)}f(e)\). We say \(G\) is weighted-\(k\)-antimagic if for any vertex weight function \(w: V\to \mathbb N\), there is an injection \(f: E\to \{1, 2, \dots , |E| + k\}\) such that for any two distinct vertices \(u\) and \(v\), \(\sum_{e\in E(v)}f(e)+w(v)\neq\sum_{e\in E(u)}f(e)+w(u)\).
A well-known conjecture asserts that every connected graph \(G\neq K_{2}\) is 0-antimagic. On the other hand, there are connected graphs \(G\neq K_{2}\) which are not weighted-1-antimagic. It is unknown whether every connected graph \(G\neq K_{2}\) 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 \(G\neq K_{2}\) on \(n\) vertices is weighted-\(\lfloor 3n/2\rfloor \)-antimagic.

MSC:
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C22 Signed and weighted graphs
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