A tour of general Hanoi graphs

The Tower of Hanoi puzzle has fascinated researchers in mathematics and theoretical computer science for over a hundred years. Many variants of the classical puzzle have been posed, such as allowing more than 3 pegs, and adding restrictions on the possible moves between the pegs. It is natural to view the pegs and allowed moves by means of a graph. The pegs are represented by the vertices of the graph, and the allowed moves by the edges. For each n, this graph yields a graph on the set of all legal configurations of n disks. Thus, the questions about the original puzzle may be reformulated as questions about connectivity and shortest paths in the graph of all configurations. Moreover, it was shown that classical Hanoi graphs are related to several interesting and useful structures such as the Sierpiński gasket and Gray codes, and thus several properties of these graphs were studied, including Hamiltonicity and planarity. In this paper we study these properties, and several others – chromatic number, clique number, and girth – for general Hanoi graphs.