arXiv subjects

Stephan Dominique Andres

Publications and source records attributed to Stephan Dominique Andres.

5 recordsLinked to original sources

arXiv2026

Counting graphs admitting a strictly matched involution

We study graphs admitting a strictly matched involution, i.e., graphs for which a strictly matched involution exists. A strictly matched involution of a graph $G$ is a self-inverse graph automorphism $f$ with the properties that the fixed point set of $f$ induces a complete subgraph in $G$ and, for the vertices that are no fixed points, every vertex is connected to its image under $f$ by an edge. Andres, Huggan, Mc Inerney, and Nowakowski (2019) determined upper and lower bounds for the number $A(n)$ of graphs admitting a strictly matched involution on $n$ vertices. We improve the upper bound and we compute the exact values for the number $A(n)$ for all $n\leq11$. Furthermore, we use combinatorial enumeration techniques and a decomposition lemma to determine the number $C(n,m)$ of graphs admitting a strictly matched involution with $n$ vertices and $m$ edges for several infinite classes of pairs $(n,m)$.

math.CO↗

arXiv2024

Line game-perfect graphs

The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $χ_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, \[χ_{[X,Y]}'(H)=ω(L(H)),\] where $ω(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.

math.CO↗

arXiv2019

A Semi-strong Perfect Digraph Theorem

Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.

math.CO↗

arXiv2017

Colouring games based on autotopisms of Latin hyper-rectangles

Every partial colouring of a Hamming graph is uniquely related to a partial Latin hyper-rectangle. In this paper we introduce the $Θ$-stabilized $(a,b)$-colouring game for Hamming graphs, a variant of the $(a,b)$-colouring game so that each move must respect a given autotopism $Θ$ of the resulting partial Latin hyper-rectangle. We examine the complexity of this variant by means of its chromatic number. We focus in particular on the bi-dimensional case, for which the game is played on the Cartesian product of two complete graphs, and also on the hypercube case.

math.CO↗

arXiv2015

The game colouring number of powers of forests

We prove that the game colouring number of the $m$-th power of a forest of maximum degree $Δ\ge3$ is bounded from above by \[\frac{(Δ-1)^m-1}{Δ-2}+2^m+1,\] which improves the best known bound by an asymptotic factor of 2.

math.CO↗