Finite groups with geodetic Cayley graphs

TL;DR

This work investigates whether finite groups can admit geodetic Cayley graphs beyond odd cycles and complete graphs. It develops structural and numerical bounds that constrain possible generating sets, enabling a comprehensive computer verification that all groups of order up to $|G|=1024$ (and many larger even-order and simple groups) satisfy the conjecture; in particular, no nontrivial geodetic counterexamples were found. The authors establish key theoretical results—such as center-based diameter and generating-set bounds, and complete-subgroup/conjugacy constraints—that apply to broad families (abelian, dihedral, large-commutativity-degree, nilpotent) and dramatically prune the search space. Their three-stage computational pipeline (filtering, preprocessing, exhaustive search) demonstrates the conjecture holds in extensive classes and provides a practical roadmap toward a potential full proof. The results underscore a deep connection between geodesic structure and group-theoretic architecture, suggesting that geodetic Cayley graphs of finite groups are extremely constrained, typically reducing to odd cycles or complete graphs.

Abstract

A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all $2$-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which significantly cuts down the number of generating sets which must be searched).

Figures (3)

• Figure 1: The situation in the proof of \ref{['lem:commgen_square']}. Note that $s,t$ have order two, so edges labeled $s,t$ are drawn as undirected.
• Figure 2: The coset $w\langle a \rangle$ in the case $w = a^k$ in the proof of \ref{['lem:commgen_coset']}.
• Figure 3: Running time of our computer search for all groups of order up to $1024$; groups of even order are marked in black, those of odd order are marked in red.

Theorems & Definitions (107)

• Conjecture 1: wrap108882
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2.1
• Lemma 2.2: stemple1968planar
• Lemma 2.3: stemple1974geodetic
• Remark 2.4
• Lemma 2.5: stemple1974geodetic
• Lemma 2.6