Optimal oriented diameter of graphs with diameter 3

Abstract

Let $f(d)$ be the smallest value for which every bridgeless graph G with diameter d admits a strong orientation $\stackrel{\rightharpoonup }{G}$ such that the diameter of $\stackrel{\rightharpoonup }{G}$ is at most $f(d)$. Chvátal and Thomassen (JCT-B, 1978) established general bounds for $f(d)$ and proved that $f(2)=6$. Kwok et al. (JCT-B, 2010) showed that $9\le f(3)\le 11$. In this paper, we determine that $f(3)=9$.

Introduction

Let $G=(V(G),E(G))$ be a finite undirected connected graph. For any $u,v\in V(G)$, the distance $d(u,v)$ is the length of a shortest path connecting u and v, and the diameter of G is defined as $d(G)=max\{d(u,v)|u,v\in V(G)\}$. A bridge of a graph G is an edge whose removal disconnects G. A graph G is bridgeless if it has no bridge. An orientation $\stackrel{\rightharpoonup }{G}$ of a graph G is a digraph obtained from it by assigning a direction to each edge. An orientation $\stackrel{\rightharpoonup }{G}$ is strong if there is a directed path from u to v for any $u,v\in V(\stackrel{\rightharpoonup }{G})$. The directed distance $\partial (u,v)$ is the length of a shortest directed path from u to v in $\stackrel{\rightharpoonup }{G}$. If $\stackrel{\rightharpoonup }{G}$ is strong, we define $d(\stackrel{\rightharpoonup }{G})=max\{\partial (u,v)|u,v\in V(\stackrel{\rightharpoonup }{G})\}$ and $\theta (u,v)=max\{\partial (u,v),\partial (v,u)\}$.

In 1939, Robbins [9] gave the well-known result that an undirected connected graph admits a strong orientation if and only if it has no bridge. Define the oriented diameter of a bridgeless graph G as:

$$\stackrel{\rightharpoonup }{diam}(G)=min\{d(\stackrel{\rightharpoonup }{G})\left|\stackrel{\rightharpoonup }{G}\text{is a strong orientation of}G\right\}.$$

If $d(\stackrel{\rightharpoonup }{G})=\stackrel{\rightharpoonup }{diam}(G)$, we call $\stackrel{\rightharpoonup }{G}$ an optimal orientation of G. Clearly, $\stackrel{\rightharpoonup }{diam}(G)\ge d(G)$, and so it is natural for us to seek an orientation $\stackrel{\rightharpoonup }{G}$ such that the difference $d(\stackrel{\rightharpoonup }{G})-d(G)$ is as small as possible. Let $f(d)$ be the smallest value for which every bridgeless graph G with diameter d admits a strong orientation $\stackrel{\rightharpoonup }{G}$ such that $d(\stackrel{\rightharpoonup }{G})\le f(d)$. This notion was introduced by Chvátal and Thomassen in [2], and they gave the following general bounds for $f(d)$.

Theorem 1

(Chvátal and Thomassen [2]) Let G be a bridgeless graph with $d(G)=d$. Then

$$\frac{1}{2}{d}^2+d\le f(d)\le 2d^2+2d.$$

Recently, Babu et al. [1] improved the upper bound in Theorem 1 to $1.373d^2+6.91d-1$ which is smaller than $2d^2+2d$ when $d\ge 8$. Furthermore, researchers have tried to obtain sharp upper bounds for some special classes of graphs such as complete k-partite graphs, or to establish tight upper bounds in terms of other graph parameters such as the domination number, minimum degree, maximum degree and so on. For example, Šoltés [10] investigated the oriented diameter of complete bipartite graphs. Gutin [4] and Plesník [8] studied the oriented diameter of complete k-partite graphs for $k\ge 3$. Their results showed that the oriented diameter is always between 2 and 4 for any complete multipartite graph. For a survey on earlier results on this topic, see [5]. In recent years, Kurz and Lätsch [6] proved that $\stackrel{\rightharpoonup }{diam}(G)\le 4\gamma (G)$, where $\gamma (G)$ is the domination number of G. For a connected bridgeless graph G of order n with maximum degree $\mathrm{\Delta }(G)=\mathrm{\Delta }$ and minimum degree $\delta (G)=\delta$, Dankelmann et al. [3] showed that $\stackrel{\rightharpoonup }{diam}(G)\le n-\mathrm{\Delta }+3$ and the bound is sharp, Surmacs [11] proved that $\stackrel{\rightharpoonup }{diam}(G)<\frac{7n}{\delta +1}$. Wang et al. [12] showed that $\stackrel{\rightharpoonup }{diam}(G)\le \lceil \frac{n}{2}\rceil$ for any maximal outplanar graph of order $n\ge 3$ with four exceptions and the bound is tight.

If $d(G)=2$, then $4\le f(2)\le 12$ by Theorem 1. However, Chvátal and Thomassen proved that $f(2)=6$ in the same paper. If $d(G)=3$, then by Theorem 1, $8\le f(3)\le 24$. In 2010, Kwok et al. [7] narrowed the gap between the upper and the lower bounds, and obtained the following.

Theorem 2

(Kwok et al. [7]) $9\le f(3)\le 11$.

The lower bound $f(3)\ge 9$ is proved in [7] by a graph on 10 vertices which is a subdivision of a $K_4$, and two optimal orientations of the graph are shown in Fig. 1.

If $d(G)=4$, then the upper bound for $f(4)$ is 40 by Theorem 1. Babu et al. [1] improved this bound to 21.

In this paper, we determine the exact value of $f(3)$ and the main result is as below.

Theorem 3

$f(3)=9$.

In the remainder of this section, we give some notations that will be used throughout this paper. For a vertex $v\in V(G)$, $N(v)=\{u|uv\in E(G)\}$, and if $U\subseteq V(G)$, then $N(U)={\cup }_{u\in U}N(u)-U$. For $S\subseteq V(G)$, $G[S]$ denotes the subgraph induced by S in G. Let U and V be two vertex sets. We use $[U,V]$ to denote the set of all edges with one endpoint in U and another in V. If $U=\{u\}$ or $V=\{v\}$, write $[u,V]$ or $[U,v]$ for short. The notation $U\to V$ means all edges in $[U,V]$ are oriented from U to V. Write $u\to V$ for $\{u\}\to V$ and $U\to v$ for $U\to \{v\}$. Let $u\in V(G)$ and $V\subseteq V(G)$ with $u\notin V$. A $(u,V)$-path is a shortest path with an endpoint u and another one in V, and $d(u,V)$ is the length of a $(u,V)$-path. For a strong orientation $\stackrel{\rightharpoonup }{G}$, a directed $(u,V)$-path is a shortest directed path from u to V, and $\partial (u,V)$ is the length of a directed $(u,V)$-path. A directed $(V,u)$-path and $\partial (V,u)$ are analogous. Because $\partial (u,V)$ and $\partial (V,u)$ may be different, we set $\theta (u,V)=max\{\partial (u,V),\partial (V,u)\}$.