Abstract
Given a simple graph $G$ with $m$ edges, we are looking for a bijection $f$ from $E(G)$ to the integer set $\{ k+1,k+2,\ldots,k+m \}$ such that the vertex sum of each vertex $v$, $\phi(v)$, defined as the sum of $f(e)$ over all edges $e$ incident to $v$ is unique. If such a bijection $f$ exists, we say $G$ is $k$-shifted antimagic. This is a generalization of the antimagic graphs proposed by Hartsfield and Ringel [7]. In this paper, we proved that every tree of diameter four or five, except for two previous known examples, is $k$-shifted antimagic for every integer $k$.
Funding Statement
The first author was supported by MOST 110-2115-M-005-005-MY2.
Acknowledgments
The authors would like to thank Prof. Zhishi Pan for many useful discussions and suggestions. Also, the authors would like to thank the anonymous referees for pointing out several typographical errors.
Citation
Wei-Tian Li. Yi-Shun Wang. "Labeling Trees of Small Diameters with Consecutive Integers." Taiwanese J. Math. 27 (3) 417 - 439, June, 2023. https://doi.org/10.11650/tjm/221103
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