A Hamilton Cycle in the $k$-Sided Pancake Network

TL;DR

This paper constructs a Hamilton cycle in the k-sided pancake network, a graph of coloured permutations, and provides algorithms for traversing this cycle, with implications for combinatorial optimization and network design.

Contribution

It introduces a Hamilton cycle in the k-sided pancake network and develops four algorithms for cycle traversal, advancing combinatorial and network theory.

Findings

Constructed a Hamilton cycle in the k-sided pancake network.

Developed four algorithms for cycle traversal.

Average flip length of cycle edges is bounded by a constant.

Abstract

We present a Hamilton cycle in the $k$-sided pancake network and four combinatorial algorithms to traverse the cycle. The network's vertices are coloured permutations $\pi = p_1p_2\cdots p_n$, where each $p_i$ has an associated colour in $\{0,1,\ldots, k{-}1\}$. There is a directed edge $(\pi_1,\pi_2)$ if $\pi_2$ can be obtained from $\pi_1$ by a "flip" of length $j$, which reverses the first $j$ elements and increments their colour modulo $k$. Our particular cycle is created using a greedy min-flip strategy, and the average flip length of the edges we use is bounded by a constant.