arXiv subjects

Adrian Rettich

Publications and source records attributed to Adrian Rettich.

3 recordsLinked to original sources

arXiv2026

Projective Chromatic Numbers

We extend classical notions of definable colourability of graphs to the general projective setting and investigate whether known results, mainly about the $G_0$ dichotomy and the $2n + 1$ conjecture, hold in the context of higher projective pointclasses. We establish that for $n \ge 2$, the presence of a $\mathbf{\Delta}^1_n$-definable well-order of the reals implies $\chi_{\mathbf{\Delta^1_n}}(G) = \chi(G)$ for all locally countable $\mathbf{\Delta^1_n}$-definable graphs $G$, and that the presence of a $\mathbf{\Delta^1_2}$-definable well-order of the reals implies $\chi_{\mathbf{\Delta^1_2}}(G) = \chi(G)$ for all locally countable Borel graphs $G$.

math.LO↗

arXiv2024

Roman Domination on Graphings

We study a variant of domination, called Roman domination, where we must assign to each vertex one of the labels 0, 1, or 2 and require that every vertex with label 0 has a neighbour with label 2. We study the problem of finding a low-cost Roman dominating function on Lebesgue-measurable graphings, that is, on infinite graphs whose vertices are the points of a probability space. We provide a framework to tackle optimisation problems in the measurable combinatorial setting. In particular, we fully answer the Roman domination problem on irrational cycle graphs, a specific type of graphing on the space $\mathbb{R}/\mathbb{Z}$ where an irrational number $α$ is given and two vertices are adjacent if and only if their distance is $α$.

math.CO↗

arXiv2024

Courcelle's Theorem: A Self-Contained Proof and a Path-Width Variant

Courcelle's Theorem is an important result in graph theory, proving the existence of linear-time algorithms for many decision problems on graphs whose tree-width is bounded by a constant. The purpose of this text is twofold: to provide an explanation and step-by-step proof of Courcelle's Theorem as applied to graphs of tree-width bounded by a constant, and to show explicitly (on the example of path-width) how to apply the same principles to other graph classes. We present these topics in a way that does not assume any particular knowledge on the part of the reader except a basic understanding of mathematics and possibly the fundamentals of graph theory. Our hope is to make the topic accessible to a broader mathematical audience, to which end we have included extensive explanations and pretty pictures.

math.CO↗