On optimal orientation of cycle vertex multiplications

Abstract

For a bridgeless connected graph G, let $\mathcal{D}(G)$ be the family of its strong orientations; and for any $D\in \mathcal{D}(G)$, we denote by $d(D)$ its diameter. The orientation number $\overrightarrow{d}(G)$ of G is defined by $\overrightarrow{d}(G)=\min \{d(D)|D\in \mathcal{D}(G)\}$. For a connected graph G of order n and for any sequence of n positive integers $(s_i)$, let $G(s_1,s_2,\dots ,s_n)$ denote the graph with vertex set $V^{*}$ and edge set $E^{*}$ such that $V^{*}={\bigcup }_{i=1}^n{V}_i$, where $V_i$'s are pairwise disjoint sets with $\left|V_i\right|=s_i$, $i=1,2,\dots ,n$, and for any two distinct vertices x, y in $V^{*}$, $\mathit{xy}\in E^{*}$ if and only if $x\in V_i$ and $y\in V_j$ for some $i,j\in \{1,2,\dots ,n\}$ with $i\ne j$ such that $v_i{v}_j\in E(G)$. We call the graph $G(s_1,s_2,\dots ,s_n)$ a G vertex multiplication. In this paper, we determine the orientation numbers of various cycle vertex multiplications.