Uroš Milutinović

Graphs

For any n ≥ 1 and any k ≥ 1, a graph S(n, k) is introduced. Vertices of S(n, k) are n-tuples over {1, 2,. . . k} and two n-tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S(n, 3) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S(n, k). Together with a formula for the distance, this result is used to compute the distance between two vertices in O(n) time. It is also shown that for k ≥ 3, the graphs S(n, k) are Hamiltonian.

S(n,k)

Sierpinski graphs S ( n , κ) generalise the Tower of Hanoi graphs—the graph S ( n , 3) is isomorphic to the graph H n 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 H n . An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to H n .

and a variant of the Tower of Hanoi problem

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 Sterns 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

It is proved that seven different approaches to the multi-peg Tower of Hanoi problem are all equivalent. Among them the classical approaches of Stewart and Frame from 1941 can be found.

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

Abstract. Several different approaches to the multi-peg Tower of Hanoi problem are equivalent. One of them is Stewarts recursive formula ¶¶

On the Frame-Stewart algorithm for the multi-peg Tower of Hanoi problem

S (n, p) = min \{2S (n_1, p) + S (n-n_1, p-1)\mid n_1, n-n_1 \in \mathbb{Z}^+\}.

Simple Explicit Formulas for the Frame-Stewart Numbers

¶¶In the present paper we significantly simplify the explicit calculation of the Frame-Stewarts numbers S(n, p) and give a short proof of the domain theorem that describes the set of all pairs (n, n1), such that the above minima are achieved at n1.

Strong products of Kneser graphs

Abstract Let G ⊠ H be the strong product of graphs G and H. We give a short proof that χ(G ⊠ H) ⩾χ(G)+2ω(H)−2 . Kneser graphs are then used to demonstrate that this lower bound is sharp. We also prove that for every n⩾2 there is an infinite sequence of pairs of graphs G and G′ such that G′ is not a retract of G while G′ ⊠ K n is a retract of G ⊠K n .

A universal separable metric space based on the triangular Sierpiński curve

Let Σ(3) be the triangular Sierpinski curve. Call the vertices of the triangles obtained during the construction of Σ(3) (with the exception of the first triangle) the rational points of Σ(3), and all other points the irrational points of Σ(3). Using results of Lipscomb [Trans. Amer. Math. Soc. 211 (1975) 143–160] and techniques and results of Milutinovic [Ph.D. thesis, 1993], [Glas. Mat. Ser. III 27 (47) (1992) 343–364], we prove that Ln(3)={x∈Σ(3)n+1:at least one coordinate of x is irrational} is a universal space for all separable metrizable spaces of dimension ⩽n.

The Classical Tower of Hanoi

This chapter describes the classical TH with three pegs. In the first section, the original task to transfer a tower from one peg to another is studied in detail. We then extend our considerations to tasks that transfer discs from an arbitrary regular state to a selected peg. We further broaden our view in Section 2.4 to tasks transforming an arbitrary regular state into another regular state. For this purpose it will be useful to introduce Hanoi graphs in Section 2.3.