Let $\Gamma $ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \Gamma $, a domatic $\aleph _0$-partition (for its Schreier graph on $\Gamma $) is a partial function $f:\Gamma \rightharpoonup \mathbb {N}$ such that for every $x\in \Gamma $, one has $f[S\cdot x]=\mathbb {N}$. We show that a continuous domatic $\aleph _0$-partition exists, if and only if a Baire measurable domatic $\aleph _0$-partition exists, if and only if the topological closure of S is uncountable. A Haar measurable domatic $\aleph _0$-partition exists for all choices of S. We also investigate domatic partitions in the general descriptive graph combinatorial setting.

The minimum roman dominating problem (denoted by γR(G),the weight of minimum roman dominating function of graph G) is a variant of the verywell known minimum dominating set problem (denoted by γ(G), thecardinality of minimum dominating set of graph G). Both problems remain NP-Complete when restrictedto P5-free graph class [A.A. Bertossi,Inf. Process. Lett. 19 (1984) 37–40; E.J. Cockayne,et al. Discret. Math. 278 (2004) 11–22]. In this paper westudy both problems restricted to some subclasses of P5-free graphs.We describe robust algorithms that solve both problems restricted to (P5,(s,t)-net)-free graphsin polynomial time. This result generalizes previous works for both problems, and improvesexisting algorithms when restricted to certain families such as (P5,bull)-freegraphs. It turns out that the same approach also serves to solve problems for generalgraphs in polynomial time whenever γ(G) and γR(G)are fixed (more efficiently than naive algorithms). Moreover, the algorithms described areextremely simple which makes them useful for practical purposes, and as we show in thelast section it allows to simplify algorithms for significant classes such ascographs.

Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$ , $\Gamma \left( R \right)$ , with vertices as elements of $R$ , where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra\,+\,Rb\,=\,R$ . In this paper, we consider a subgraph ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$ , where $\text{J}\left( R \right)$ is the Jacobson radical of $R$ . We associate the ring properties of $R$ , the graph properties of ${{\Gamma }_{2}}\left( R \right)$ , and the topological properties of $\text{Max}\left( R \right)$ . Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, …, X n }, where X 1, X 2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .