Graph-theoretical estimates of the diameters of the 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 Bollobás and de la Vega. They range from 60% to 77% of the correct diameters of large-$n$ graphs.

Figures (6)

• Figure 1: Generalized Petersen graph $G(4,1)$ as the Cayley graph of the $3\times3\times3$ Rubik's Cube group in the square-slice-turn metric. Each vertex is a unique configuration (a pattern of face colors of the cubies while holding the orientation of the whole Cube fixed) reachable by a combination of the $R^2L^{-2}$, $D^2U^{-2}$, and $B^2F^{-2}$ turns (in the Singmaster notation Singmasterbook) from the solved configuration. Each edge connecting two vertexes is a turn (and its inverse) that brings one configuration denoted by one of the vertexes to the other. The order of the graph is 8, the degree is 3, the girth ($g$) is 4, the diameter is 3, and the number of $g$-cycles traversing a vertex is 3.
• Figure 2: Generalized Petersen graph $G(12,5)$ as the Cayley graph of the $2\times2\times2$ Rubik's Cube group in the square-turn metric. The order is 24, the degree is 3, the girth ($g$) is 6, the diameter is 4, and the number of $g$-cycles per vertex is 3.
• Figure 3: A partial Cayley graph of the 2$\times$2$\times$2 Rubik's Cube group in the quarter-turn metric, which is symmetric with degree 6. The order is $3.67 \times 10^6$, the girth ($g$) is 4, the diameter is 14, and the number of $g$-cycles per vertex is 3. The vertexes on the periphery are further connected to other vertexes that are not shown.
• Figure 4: The distance array $\bm{d}$ of the locally isomorphic Cayley graphs of the 2$\times$2$\times$2 Rubik's Cube group in the quarter-turn metric and of the 3$\times$3$\times$3 Rubik's Cube group in the square-turn metric. The terminal point in each curve corresponds to the diameter.
• Figure 5: A partial Cayley graph of the 3$\times$3$\times$3 Rubik's Cube group in the square-turn metric, which is symmetric with degree 6. The order is $663\,552$, the girth ($g$) is 4, the diameter is 15, and the number of $g$-cycles per vertex is 3.