Optimal orientations of Vertex-multiplications of Trees with Diameter 4

Authors

Willie Han Wah Wong, National Institute of Education, Nanyang Technological University of SingaporeFollow
Eng Guan Tay, National Institute of Education, Nanyang Technological University of SingaporeFollow

Abstract

Koh and Tay proved a fundamental classification of G vertex-multiplications into three classes ζ₀, ζ₁ and ζ₂. They also showed that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class ζ₂. 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 ζ₀ (or ζ₁) 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 ≥ 3.

Creative Commons License

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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