Isometric Cycles and a Generalization of Moore Graphs

TL;DR

The paper introduces the equator $eqt(G)$ as the length of the longest isometric cycle and derives a Moore-type lower bound on the order $n$ in terms of the equator $q$, girth $g$, and minimum degree $\delta$, with $n \ge \frac{q}{g} M(\delta,g)$ for $q$ sufficiently large. It then identifies and studies equatorial graphs that attain this bound, proving regularity, a unique canonical partition, and a structural framework that generalizes Moore graphs. The authors completely characterize equatorial graphs for small girth ($g=3,4$) and the $(3,5)$-case, and they establish sharpness results via Moore graphs, cages, and Brown graphs for the $C_4$-free bound. The work also develops existence results for infinite families of equatorial graphs and proposes a conjecture linking their existence to Moore graphs, with several open questions about $\,(\delta,g,q)\,$-cages. Overall, the study extends classical Moore bounds to a broader equator-based setting and reveals deep structural parallels to cages and Moore graphs.

Abstract

The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $δ$ - and show that this bound is sharp when there exists a Moore graph with girth $g$ and minimum degree $δ$. The extremal graphs that attain our bound give an analogue of Moore graphs. We prove that these extremal `Moore-like' graphs are regular, and that every one of their vertices is contained in some maximum length isometric cycle. We show that these extremal graphs have a highly structured partition that is unique, and easily derived from any of its maximum length isometric cycles. We characterize the extremal graphs with girth 3 and 4, and those with girth 5 and minimum degree 3. We also bound the order of $C_4$-free graphs with given equator and minimum degree, and show that this bound is nearly sharp. We conclude with some questions and conjectures further relating our extremal graphs to cages and Moore graphs.

Figures (4)

• Figure 1: Left is a breadth-first search tree rooted at the bold vertex in a $(3,5)$-graph, and right is one rooted at the bold edge in a $(3,6)$-graph.
• Figure 2: On the left is a subgraph of the graph $\mathcal{F}(4, 3, q)$ constructed in the proof of Theorem \ref{['thm:sharp_moore_parameters']}. On the right is a subgraph of $\mathcal{F}(3, 4, q)$. In both diagrams, the vertices $u_i$ are black, and the vertices $v_i$ are grey.
• Figure 3: Copies of the gadget $H_i$ above are used to build the $C_4$-free graph $\mathcal{H}(\delta, q)$.
• Figure 4: The equatorial graph $\mathcal{F}(3,5,20)$, considered as a reflexive graph with loops on every vertex, retracts onto the bolded isometric cycle. If two vertices are in the same part $L_i$ of the partition induced by isometric cycles, then they are drawn the same color.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4