Hanoi Graphs Research Articles

The diameter of Hanoi graphs

Many questions regarding the Tower of Hanoi problem have been posed and answered during the years. Variants of the classical puzzle, such as allowing more than 3 pegs, and imposing limitations on the possible moves among the pegs, raised the analogous questions for those variants. One such question is: given a variant, and a certain number of disks, find a pair of disk arrangements such that the minimal number of moves required for changing from the first to the second is maximal over all pairs. One of the main results of the paper is identifying these for the Cyclic h variants—the variants with h pegs arranged along a uni-directional circle—to be the pairs of perfect configurations where the destination peg is right before the source peg.

Shortest Paths in the Tower of Hanoi Graph and Finite Automata

We present efficient algorithms for constructing a shortest path between two configurations in the Tower of Hanoi graph and for computing the length of the shortest path. The key element is a finite-state machine which decides, after examining on the average only a small number of the largest discs (asymptotically, $\frac{63}{38} \approx 1.66$), whether the largest disc will be moved once or twice. This solves a problem raised by Andreas Hinz and results in a better understanding of how the shortest path is determined. Our algorithm for computing the length of the shortest path is typically about twice as fast as the existing algorithm. We also use our results to give a new derivation of the average distance $\frac{466}{885}$ between two random points on the Sierpin´ski gasket of unit side.

CODES AND $L(2,1)$-LABELINGS IN SIERPIŃSKI GRAPHS

The $\lambda$-number of a graph $G$ is the minimum value $\lambda$ such that $G$ admits a labeling with labels from $\{0,1,\ldots,\lambda\}$ where vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. Sierpiński graphs $S(n,k)$ generalize the Tower of Hanoi graphs---the graph $S(n,3)$ is isomorphic to the graph of the Tower of Hanoi with $n$ disks. It is proved that for any $n\geq 2$ and any $k \geq 3$, $\lambda(S(n,k)) = 2k$. To obtain the result (perfect) codes in Sierpiński graphs are studied in detail. In particular a new proof of their (essential) uniqueness is obtained.

Hanoi graphs and some classical numbers

The Hanoi graphs H p n model the p-pegs n-discs Tower of Hanoi problem(s). It was previously known that Stirling numbers of the second kind and Stern's diatomic sequence appear naturally in the graphs H p n . In this note, second-order Eulerian numbers and Lah numbers are added to this list. Considering a variant of the p-pegs n-discs problem, Catalan numbers are also encountered.

CODES AND $L(2,1)$-LABELINGS IN SIERPI\'NSKI GRAPHS

The $\lambda$-number of a graph $G$ is the minimum value $\lambda$ such that $G$ admits a labeling with labels from $\{0,1,\ldots, \lambda\}$ where vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. Sierpi\'nski graphs $S(n,k)$ generalize the Tower of Hanoi graphs---the graph $S(n,3)$ is isomorphic to the graph of the Tower of Hanoi with $n$ disks. It is proved that for any $n\geq 2$ and any $k\geq 3$, $\lambda(S(n,k)) = 2k$. To obtain the result (perfect) codes in Sierpi\'nski graphs are studied in detail. In particular a new proof of their (essential) uniqueness is obtained.

Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence

It is known that in the Tower of Hanoi graphs there are at most two different shortest paths between any fixed pair of vertices. A formula is given that counts, for a given vertex v, the number of vertices u such that there are two shortest u, v-paths. The formula is expressed in terms of Stern’s diatomic sequence b( n) ( n≥0) and implies that only for vertices of degree two this number is zero. Plane embeddings of the Tower of Hanoi graphs are also presented that provide an explicit description of b( n) as the number of elements of the sets of vertices of the Tower of Hanoi graphs intersected by certain lines in the plane.

1-perfect codes in Sierpiński graphs

Sierpiński graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.

On the planarity of Hanoi graphs

Hanoi graphs are the state graphs for Tower of Hanoi problems with three or more pegs. We prove hamiltonicity and present a complete analysis of planarity of these graphs.

Error-correcting codes on the towers of Hanoi graphs

A perfect one-error-correcting code on a graph is a subset of the vertices so that no two vertices in the subset are adjacent and each vertex not in the subset is adjacent to exactly one vertex in the subset. We show that the Towers of Hanoi puzzle defines an infinite family of graphs, and that each such graph supports a perfect one-error-correcting code. We show that these codes are essentially unique. Our characterization of the codewords as those ternary strings with an even number of 1's and an even number of 2's, makes generation and decoding computationally easy. In particular, decoding can be carried out by a two-pass finite state machine. We also show that determining if a graph can support a perfect one-error-correcting code is an NP-complete problem.

Perfect codes on the towers of Hanoi graph

We characterise all the perfect k-error correcting codes that can be defined on the graph associated with the Towers of Hanoi puzzle. In particular, a short proof for the existence of 1-error correcting code on such a graph is given.

A statistical analysis of the towers of hanoi problem

The Towers of Hanoi problem is analyzed with the aid of Hanoi graphs depicting legal configurations (i. e. arrangements of the disks on the pegs) and the transitions among them. Specifically, the mean and the standard deviation of the minimum numbers of moves required to change an arbitrary initial configuration into an arbitrary final configuration are computed, and are shown to be approximately 0.52655 · 2 n and 0.25025 · 2 n respectively for n disks.

Towers of hanoi graphs

The notion of Hanoi Graph is introduced the first time for the Towers of Hanoi puzzle which greatly helps investigate the problem. And some new conclusions are presented in this paper.