A Stratification of the Hanoi Graph for 4 Pegs

For the Tower of Hanoi problem, it has been found that it is possible to construct a nice graph with legal states of the Tower of Hanoi as the vertices and legal moves between legal states as the edges between corresponding vertices. This “Hanoi graph” we denote by Hk n for k pegs and n disks. Our motivation is to describe the properties of this graph for k ≥ 4 and to describe a class of graphs called Stratified Hanoi graphs, appropriately denoted by SHk n, which will ultimately be useful in visualizing the FrameStewart algorithm. 1. Background Information In this article, we will follow the convention of [POOLE] to label vertices. Our motivation is to describe a class of graphs called Hanoi graphs, denoted by H n for k pegs and n disks, and to ultimately describe Stratified Hanoi graphs, appropriately denoted by SH n for k pegs and n disks. In order to do that, in this section we will introduce some notation and definitions that we will use throughout this article. Definition 1.1. A legal state is a configuration of n disks on the k pegs in which each disk is either on top of a larger disk or on an empty peg. Let us label the pegs 0, 1, · · · , k − 1 and the disks from smallest to largest 0, 1, · · · , n−1. We label each legal state with an n-bit k-ary string an−1an−2...a0 where ai ∈ {0, 1, · · · , k − 1} and ai = j if disk i lies on peg j. It is not difficult to see how this labeling is useful, because each string corresponds to a unique legal state. In this article, we will denote the vertex set of a graph G by V (G) and the edge set of G by E(G). Definition 1.2. A legal move from one legal state to another legal state is achieved by moving exactly one disk from one peg to another peg under the following rules. (1) A disk can be moved onto a larger disk. (2) A disk can be moved onto an empty peg. (3) A disk cannot be moved onto a smaller disk. Definition 1.3. Let k,n be positive integers. We define H n as a graph such that V (H n) is the set of vertices each labeled by an n-bit k-ary string which corresponds to a unique legal state, and E(H n) is the set of edges each of which can only exist between two vertices if there exists a legal move between their corresponding legal states. We can see that |V (H n)| = k. Date: July 31, 2008.