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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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