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Weighted-1-antimagic graphs of prime power order. (English) Zbl 1244.05186
Summary: Suppose $$G$$ is a graph, $$k$$ is a non-negative integer. We say $$G$$ is weighted-$$k$$-antimagic if for any vertex weight function $$w:V\to \mathbb N$$, there is an injection $$f:E\to \{1,2,\dots ,\mid E\mid +k\}$$ such that for any two distinct vertices $$u$$ and $$v$$,
$\sum _{e\in E(v)}f(e)+w(v)\neq \sum _{e\in E(u)}f(e)+w(u).$ There are connected graphs $$G\neq K_{2}$$ which are not weighted-1-antimagic.
It was asked in T. Wong and X. Zhu [J. Graph Theory 70, No. 3, 348–350 (2012; Zbl 1244.05192)] whether every connected graph other than $$K_{2}$$ is weighted-2-antimagic, and whether every connected graph on an odd number of vertices is weighted-1-antimagic. It was proved in T. Wong and X. Zhu [loc. cit] that if a connected graph $$G$$ has a universal vertex, then $$G$$ is weighted-2-antimagic, and moreover if $$G$$ has an odd number of vertices, then $$G$$ is weighted-1-antimagic.
In this paper, by restricting to graphs of odd prime power order, we improve this result in two directions: if $$G$$ has odd prime power order $$p^{z}$$ and has total domination number 2 with the degree of one vertex in the total dominating set not a multiple of $$p$$, then $$G$$ is weighted-1-antimagic. If $$G$$ has odd prime power order $$p^{z}$$, $$p \neq 3$$ and has maximum degree at least $$|V(G)| - 3$$, then $$G$$ is weighted-1-antimagic.

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