Cycles in the burnt pancake graphs

TL;DR

This paper characterizes all cycle lengths in burnt pancake graphs, showing they contain cycles of all lengths from 8 up to their maximum, and provides a detailed description of the smallest cycles.

Contribution

It proves that burnt pancake graphs contain cycles of all lengths from 8 to their maximum and characterizes all 8-cycles explicitly.

Findings

Burnt pancake graphs have cycles of all lengths from 8 to 2^n n!

Complete characterization of all 8-cycles in BP_n for n ≥ 2

Constructive proof using recursive structure of BP_n

Abstract

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.