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arXiv · 2001.01908

On optimal orientations of complete tripartite graphs

W. H. W. Wong·E. G. Tay
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Abstract

Given a connected and bridgeless graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$. The orientation number of $G$ is defined to be $\bar{d}(G):=min\{d(D)|D\in \mathscr{D}(G)\}$, where $d(D)$ is the diameter of the digraph $D$. In this paper, we focus on the orientation number of complete tripartite graphs. We prove a conjecture raised by Rajasekaran and Sampathkumar. Specifically, for $q\ge p\ge 3$, if $\bar{d}(K(2,p,q))=2$, then $q\le{{p}\choose{\lfloor{p/2}\rfloor}}$. We also present some sufficient conditions on $p$ and $q$ for $\bar{d}(K(p,p,q))=2$.

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W. H. W. Wong, E. G. Tay. 2020-01-07. On optimal orientations of complete tripartite graphs. https://arxiv.org/abs/2001.01908

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