Stern polynomials B_k(t) 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. It is also proved that the degree of B_n(t) equals the difference between the length and the weight of the non-adjacent form of n.
COBISS.SI-ID: 14276441
Sierpiński graphs S(n,3) are the graphs of the Tower of Hanoi puzzle with n disks, Sierpiński gasket graphs S_n, are the graphs naturally defined by the finite number of iterations that lead to the Sierpiński gasket. An explicit labeling of the vertices of S_n, is introduced. It is proved that S_n is uniquely 3-colorable, that S(n,3) is uniquely 3-edge-colorable, and that chi'(S_n) = 4, thus answering a question from [Australas. J. Combin. 35 (2006) 181-192]. It is also shown that S_n contains a 1-perfect code only for n = 1 or n = 3 and that every S(n,3) contains a unique Hamiltonian cycle.
COBISS.SI-ID: 14677593
In 2008 our research monograph was published that covers contemporary topics from graph theory. Throughout the book the topics are interrelated via the Cartesian product of graphs. Of course just a part of the book is devoted to the topic of this researh project. More precisely, the Tower of Hanoi graphs are presented as an important example of subgraphs of Cartesian products of complete graphs that are in turn known as Hamming graphs.
COBISS.SI-ID: 14965081
In order to find the optimal solution of a task, i.e. the shortest path between two given states of the Tower of Hanoi, it is a first step to determine the moves of the largest disc. Surprisingly, there are shortest paths with more than one largest disc move. We analyze the exsistence of shortest paths with a given pattern of largest disc moves using numerical and analytical methods.
COBISS.SI-ID: 15079513
Tower of Hanoi with three pegs is a source for several (non-repetitive) sequences. The sequences presented are the greedy non-repetititve sequence, two non-repetitive sequences on six and five symbols, respectively, that are defined with the pairs of pegs involved in the steps of the optimal solution, and the Prouhet-Thue-Morse sequence. Some open problems are also listed.
COBISS.SI-ID: 15071833