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

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

The discovery of two passages from 1769 by the German Georg Christoph Lichtenberg and the Japanese Yoriyuki Arima, respectively, sheds some new light on the early history of integer sequences and mathematical induction. Both authors deal with the solution of the ancient Chinese rings puzzle, where metal rings are moved up and down on a very sophisticated mechanical arrangement. They obtain the number of (necessary) moves to solve it in the presence of n rings. While Lichtenberg considers all moves, Arima concentrates on the down moves only of the first ring. We will present a unified view on integer sequences and discuss some of their most fundamental representatives before collecting properties of the Lichtenberg sequence $\ell_n$, defined mathematically by the recurrence $\ell_n + \ell_{n-1} = 2n - 1$, and related sequences such as the Jacobsthal sequence, which is the sequence of differences of $\ell$. And, of course, at some point Fibonacci numbers will enter the scene.

COBISS.SI-ID: 17955673