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… (More)
It is proved that seven diierent 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.
In contrast to the widespread interest in the Frame-Stewart Conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs H n p in a more general setting. Here p stands for the number of pegs and n for the number of… (More)
Sierpi nski graphs S(n; k) generalize 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 eecient dominating set) in a graph G is a vertex subset of G with the property that the closed neighborhoods of its elements form a partition of V (G). It is proved that the graphs S (n;… (More)
Combinatorial properties of the multi-peg Tower of Hanoi problem on n discs and p pegs are studied. Top-maps are introduced as maps which reflect topmost discs of regular states. We study these maps from several points of view. We also count the number of edges in graphs of the multi-peg Tower of Hanoi problem and in this way obtain some combinatorial… (More)
Stern polynomials B k (t), k ≥ 0, t ∈ R, are introduced in the following way: B 0 (t) = 0, B 1 (t) = 1, B 2n (t) = tB n (t), and B 2n+1 (t) = B n+1 (t) + B n (t). It is shown that B n (t) has a simple explicit representation in terms of the hyperbinary representations of n − 1 and that B ′ 2n−1 (0) equals the number of 1's in the standard Gray code for n−1.… (More)
The Hanoi graphs H n p model the p-pegs n-discs Tower of Hanoi problem(s). It was previously 9 known that Stirling numbers of the second kind and Stern's diatomic sequence appear naturally in the graphs H n p. In this note, second-order Eulerian numbers and Lah numbers are added to this list. 11 Considering a variant of the p-pegs n-discs problem, Catalan… (More)