Home > Journals > TAG > Vol. 10 > Iss. 1 (2023)
Abstract
\noindent Koh and Tay proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. They also showed that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Of interest, $G$ vertex-multiplications are extensions of complete $n$-partite graphs and Gutin characterised complete bipartite graphs with orientation number 3 (or 4 resp.) via an ingenious use of Sperner's theorem. In this paper, we investigate vertex-multiplications of trees with diameter $4$ in $\mathscr{C}_0$ (or $\mathscr{C}_1$) and exhibit its intricate connections with problems in Sperner Theory, thereby extending Gutin's approach. Let $s$ denote the vertex-multiplication of the central vertex. We almost completely characterise the case of even $s$ and give a complete characterisation for the case of odd $s\ge 3$.
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Recommended Citation
Wong, Willie Han Wah and Tay, Eng Guan
(2023)
"Optimal orientations of Vertex-multiplications of Trees with Diameter 4,"
Theory and Applications of Graphs: Vol. 10:
Iss.
1, Article 6.
DOI: 10.20429/tag.2023.10106
Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol10/iss1/6