The girths of the cubic Pancake graphs

TL;DR

This paper investigates the girth of cubic Pancake graphs, which are specific Cayley graphs generated by prefix-reversals, and establishes an upper bound of twelve for their girth.

Contribution

The paper provides the first analysis of the girth of cubic Pancake graphs, showing they have girths at most twelve, advancing understanding of their structural properties.

Findings

Cubic Pancake graphs have girth at most twelve.

Six generating sets of prefix-reversals produce connected Cayley graphs.

The girth bounds contribute to the structural characterization of these graphs.

Abstract

The Pancake graphs $P_n, n\geqslant 2$, are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by prefix-reversals. There are six generating sets of prefix-reversals of cardinality three which give connected Cayley graphs over the symmetric group known as cubic Pancake graphs. In this paper we study the girth of the cubic Pancake graphs. It is proved that considered cubic Pancake graphs have the girths at most twelve.