Domination and upper domination of direct product graphs

Let $X_{\mathbb{Z}∕n\mathbb{Z}}$ denote the unitary Cayley graph of $\mathbb{Z}∕n\mathbb{Z}$. We present results on the tightness of the known inequality $\gamma \left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le {\gamma }_t\left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le g\left(n\right)$, where $\gamma$ and${\gamma }_t$ denote the domination number and total domination number, respectively, and $g$ is the arithmetic function known as Jacobsthal’s function. In particular, we construct integers $n$ with arbitrarily many distinct prime factors such that $\gamma \left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le {\gamma }_t\left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le g\left(n\right)?1$. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.     

Discrete Tomography of Mathematcal Quasicrystals: A Primer

This text is a report on work progress. We introduce the class of cyclotomic model sets (mathematical quasicrystals) $\Lambda \subset \mathbb{Z}[{\xi }_n]$, where $\mathbb{Z}[{\xi }_n]$ is the ring of integers in the nth cyclotomic field $\mathbb{Q}({\xi }_n)$, and discuss the corresponding decomposition, consistency and reconstruction problems of the discrete tomography of these sets. Our solution of the so-called decomposition problem also applies to the case of the square lattice ${\mathbb{Z}}^2=\mathbb{Z}[{\xi }_4]$, which corresponds to the classical setting of discrete tomography.     

Game of cops and robbers in oriented quotients of the integer grid

The integer grid $\mathbb{Z}\square \mathbb{Z}$ has four typical orientations of its edges which make it a vertex-transitive digraph. In this paper we analyze the game of Cops and Robbers on arbitrary finite quotients of these directed grids.     

The admissibility of M11 over number fields

A group G is $\mathbb{Q}$-admissible if there exists a G-crossed product division algebra over $\mathbb{Q}$. The $\mathbb{Q}$-admissibility conjecture asserts that every group with metacyclic Sylow subgroups is $\mathbb{Q}$-admissible. We prove that the Mathieu group $M_{11}$ is $\mathbb{Q}$-admissible, in contrast to any other sporadic group.