Oriented diameter of graphs with diameter $4$ and given edge girth

TL;DR

The paper addresses the problem of oriented diameter for bridgeless graphs with a fixed diameter by introducing the edge-girth parameter $g^*(G)$ and the derived function $F(d,g^*)$, proving the pivotal relation $f(d)=\\max\\{F(d,g^*): 2\\le g^*\\le 2d+1\\}$. Using constructive orientations, including the $R$-\\$S$ orientation framework and partition-based techniques, the authors determine sharp bounds for $d=4$ across several edge-girth values: $F(4,2)=4$, $F(4,9)=12$, $F(4,3)\\le 12$, and $F(4,g^*)\\le 13$ for $g^*\\in\\{6,7,8\\}$, while also showing $F(4,9)\\ge 12$. They further establish $F(4,2)=4$ and provide $F(4,9)=12$ with a lower bound via a subdivision of $K_4$, together with an exact bound for $F(4,3)$ and bounds for the remaining $g^*$ values. The results advance the understanding of oriented diameters by connecting diameter constraints with edge girth and offering a pathway to tighten the classical bounds on $f(d)$, while highlighting open questions for $f(4)$ and related $F(4,g^*)$ values.

Abstract

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chvátal and Thomassen (JCT-B, 1978) obtained general bounds for $f(d)$ and proved that $f(2)=6$. Kwok et al. (JCT-B, 2010) proved that $9\leq f(3)\leq 11$. Wang and Chen (JCT-B, 2022) determined $f(3)=9$. Babu et al. (DAM, 2021) showed $f(4)\leq 21$. In this paper, we introduce a new approach to studying $f(d)$ via the edge girth of a bridgeless graph $G$, denoted by $g^*(G)=\max\{l_G(e)\mid e\in E(G)\}$, where $l_G(e)$ is the length of the shortest cycle containing $e$ in $G$. Then we define $F(d,g^*)=\max\{\overrightarrow{diam}(G)\mid G\text{ is bridgeless},d(G)=d,g^*(G)=g^*\}$, and show $f(d)=\max\{F(d,g^*)\mid 2\leq g^*\leq 2d+1\}$. As the main result of this paper, we establish $F(4,2)=4$, $F(4,9)=12$, $F(4,3)\le 12$, and $F(4,g^*)\le 13$ for $g^*\in\{6,7,8\}$, and we propose two open problems for further research.

Figures (6)

• Figure 1: $H_{d,g^*}$ with $g^*\geq 3$ and $k=\lfloor\frac{g^*}{2}\rfloor+1$.
• Figure 2: $H_{d,g^*}$ with $g^*=2$.
• Figure 3: $R$-$S$ orientation.
• Figure 4: The bridgeless graph $G$ with $l_G(e)=g^*(G)\in\{6,7,8,9\}$.
• Figure 5: Orientations of some edges of $G$ with $g^*(G)\in \{6,7,8\}$.

Theorems & Definitions (23)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.4
• Claim 1
• Claim 2