Coloring and Counting on the Tower of Hanoi Graphs

The Tower of Hanoi graphs make up a beautifully intricate and highly symmetric family of graphs that show moves in the Tower of Hanoi puzzle played on three or more pegs. Although the size and order of these graphs grow exponentially large as a function of the number of pegs, p, and disks, d (there are pd vertices and even more edges), their chromatic number remains remarkably simple. The interplay between the puzzles and the graphs provides fertile ground for counts, alternative counts, and still more alternative counts.