arXiv subjects
Publications and source records attributed to Leo Versteegen.
We show that the maximum number of minimum dominating sets of a forest with domination number $γ$ is at most $\sqrt{5}^γ$ and construct for each $γ$ a tree with domination number $γ$ that has more than $\frac{2}{5}\sqrt{5}^γ$ minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.
We study relations between diameter $D(G)$, domination number $γ(G)$, independence number $α(G)$ and cop number $c(G)$ of a connected graph $G$, showing (i.) $c(G) \leq α(G)-\lfloor \frac{D(G)-3}{2} \rfloor$, and (ii.) $c(G) \leq γ(G) - \frac{D(G)}{3} + O (\sqrt{D(G)})$.
We present an algorithm of time complexity $O(kn^{k+2})$ deciding whether a graph $G$ on $n$ vertices is $k$-copwin. The fastest algorithm thus far had time complexity $O(n^{2k+2})$.
A linear configuration is said to be common in $G$ if every 2-coloring of $G$ yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogously to a result by Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in $\mathbb{F}_p^n$ for $p\geq 5$ and large $n$ and in $\mathbb{Z}_p$ for large primes $p$.
A linear configuration is said to be common in a finite Abelian group $G$ if for every 2-coloring of $G$ the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation over $G$, then it is common in $\mathbb{F}_p^n$ if and only if the equation's coefficients can be partitioned into pairs that sum to zero mod $p$. This was proven by Fox, Pham and Zhao for sufficiently large $n$. We generalize their result to all sufficiently large Abelian groups $G$ for which the equation's coefficients are coprime to $\vert G\vert$