The proof of a conjecture about cages

TL;DR

This paper proves a longstanding conjecture that in minimal k-regular graphs with girth g, all cycles of length g are nonseparating, extending the proof to include odd girth values.

Contribution

The paper provides a proof confirming that all cycles of length g in (k; g)-cages are nonseparating, completing the proof for both even and odd girth values.

Findings

Confirmed the conjecture for odd girth g in (k; g)-cages.

Extended previous proofs that only covered even girth g.

Established that all cycles of length g are nonseparating in these cages.

Abstract

The girth of a graph is defined as the length of a shortest cycle in the graph. A $(k; g)$-cage is a graph of minimum order among all $k$-regular graphs with girth $g$. A cycle $C$ in a graph $G$ is termed nonseparating if the graph $G-V(C)$ remains connected. A conjecture, proposed in [T. Jiang, D. Mubayi. Connectivity and Separating Sets of Cages. J. Graph Theory 29(1)(1998) 35--44], posits that every cycle of length $g$ within a $(k; g)$-cage is nonseparating. While the conjecture has been proven for even $g$ in the aforementioned work, this paper presents a proof demonstrating that the conjecture holds true for odd $g$ as well. Thus, the previously mentioned conjecture was proven to be true.