An improvement on the maximum number of $k$-Dominating Independent Sets

TL;DR

This paper investigates the maximum number of $k$-dominating independent sets in graphs, disproves a previous conjecture for even $k$, and provides improved bounds on their growth rate.

Contribution

It disproves Nagy's conjecture for even $k$ and establishes new bounds on the asymptotic growth of the maximum number of $k$-dominating independent sets.

Findings

Disproved Nagy's conjecture for even $k$.

Established lower bound $oxed{1.489}$ for $ oot n oot[k]{mi_k(n)}$.

Improved upper bound $oxed{1.98}$ for the growth rate of $mi_k(n)$.

Abstract

Erd\H{o}s and Moser raised the question of determining the maximum number of maximal cliques or equivalently, the maximum number of maximal independent sets in a graph on $n$ vertices. Since then there has been a lot of research along these lines. A $k$-dominating independent set is an independent set $D$ such that every vertex not contained in $D$ has at least $k$ neighbours in $D$. Let $mi_k(n)$ denote the maximum number of $k$-dominating independent sets in a graph on $n$ vertices, and let $\zeta_k:=\lim_{n \rightarrow \infty} \sqrt[n]{mi_k(n)}$. Nagy initiated the study of $mi_k(n)$. In this article we disprove a conjecture of Nagy and prove that for any even $k$ we have $$1.489 \approx \sqrt[9]{36} \le \zeta^k_k.$$ We also prove that for any $k \ge 3$ we have $$\zeta_k^{k} \le 2.053^{\frac{1}{1.053+1/k}}< 1.98,$$ improving the upper bound of Nagy.