Graph-theoretical estimates of the diameters of the Rubik's Cube groups

TL;DR

This paper introduces a new method to estimate the diameters of symmetric graphs, specifically applied to Rubik's Cube groups, providing tighter lower bounds that approach the actual diameters more closely than previous estimates.

Contribution

It proposes a calculable lower bound for graph diameters based on local parameters, and applies it to Rubik's Cube groups to improve diameter estimates.

Findings

Lower bounds range from 60% to 77% of actual diameters.

Method yields tighter bounds than previous random graph estimates.

Applicable to various sizes and metrics of Rubik's Cube groups.

Abstract

A strict lower bound for the diameter of a symmetric graph is proposed, which is calculable with the order $n$ and other local parameters of the graph such as the degree $k\,(\geq 3)$, even girth $g\,(\geq 4)$, and number of $g$-cycles traversing a vertex, which are easily determined by inspecting a small portion of the graph (unless the girth is large). It is applied to the symmetric Cayley graphs of some Rubik's Cube groups of various sizes and metrics, yielding slightly tighter lower bounds of the diameters than those for random $k$-regular graphs proposed by Bollob\'{a}s and de la Vega. They range from 60% to 77% of the correct diameters of large-$n$ graphs.