The solitaire game "The Tower of Hanoi" was invented in the 19th century by the French number theorist Édouard Lucas. The book presents its mathematical theory and offers a survey of the historical development from predecessors up to recent research. In addition to long-standing myths, it provides a detailed overview of the essential mathematical facts with complete proofs, and also includes unpublished material, e.g., on some captivating integer sequences. The main objects of research today are the so-called Hanoi graphs and the related Sierpiński graphs. Acknowledging the great popularity of the topic in computer science, algorithms, together with their correctness proofs, form an essential part of the book. In view of the most important practical applications, namely in physics, network theory and cognitive (neuro)psychology, the book also addresses other structures related to the Tower of Hanoi and its variants. The updated second edition includes, for the first time in English, the breakthrough reached with the solution of the "The Reve's Puzzle" in 2014. This is a special case of the famed Frame-Stewart conjecture which is still open after more than 75 years. Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike.

COBISS.SI-ID: 18363737

The purpose of this survey is to bring some order into the growing literature on a type of graphs which emerged in the past couple of decades under a wealth of names and in various disguises in different fields of mathematics and its applications. The central role is played by Sierpiński graphs, but we will also shed some light on variants of these graphs and in particular propose their classification. Concentrating on Sierpiński graphs proper we present results on their metric aspects, domination-type invariants with an emphasis on perfect codes, different colorings, and embeddings into other graphs.

COBISS.SI-ID: 17827417

If $G$ is a graph and $n$ a positive integer, then the generalized Sierpiński graph $S_G^n$ is a fractal-like graph that uses $G$ as a building block. The construction of $S_G^n$ generalizes the classical Sierpiński graphs $S_p^n$, where the role of $G$ is played by the complete graph $K_p$. An explicit formula for the number of connected components in $S_G^n$ is given and it is proved that the (edge-)connectivity of $S_G^n$ equals the (edge-)connectivity of $G$. It is demonstrated that $S_G^n$ contains a $1$-factor if and only if $G$ contains a $1$-factor. Hamiltonicity of generalized Sierpiński graphs is also discussed.

COBISS.SI-ID: 18473561

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in\{1,\ldots , k\}$, where each $V_i$ is an $i$-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs $ST_3^n$ is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs $S_G^n$ with respect to all connected graphs $G$ of order 4.

COBISS.SI-ID: 18480985

We continue the study of the Grundy domination number of a graph. A linear algorithm to determine the Grundy domination number of an interval graph is presented. The exact value of the Grundy domination number of an arbitrary Sierpiński graph is proven, and efficient algorithms to construct the corresponding sequences are presented. These results are obtained by using sharp bounds for the Grundy domination number of a vertex- and edge-removed graph, proven in this paper.

COBISS.SI-ID: 17807705