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arXiv · 2110.09003

Application of some techniques in Sperner Theory: Optimal orientations of vertex-multiplications of trees with diameter 4

W. H. W. Wong·E. G. Tay
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Abstract

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 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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W. H. W. Wong, E. G. Tay. 2021-10-18. Application of some techniques in Sperner Theory: Optimal orientations of vertex-multiplications of trees with diameter 4. https://arxiv.org/abs/2110.09003

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