On the number of maximal independent sets and maximal induced bipartite subgraphs in $K_4$-free graphs

TL;DR

This paper establishes upper bounds on the number of maximal independent sets of a given size and maximal induced bipartite subgraphs in $K_4$-free graphs, advancing understanding of their combinatorial structure.

Contribution

It provides new bounds on the counts of these subgraphs in $K_4$-free graphs, introducing constants that refine previous estimates.

Findings

Bound on maximal independent sets of size $k$ in $K_4$-free graphs

Bound on maximal induced bipartite subgraphs in $K_4$-free graphs

Existence of positive constants $ heta$ and $ u$ for these bounds

Abstract

Let $G$ be a $K_4$-free graph of order $n$ and let $k$ be an integer with $0\leq k\leq n$. We show the existence of positive constants $\eta$ and $\nu$ such that $G$ has at most $(4-\eta)^{(5-\eta)k-n}(5-\eta)^{n-(4-\eta)k}$ maximal independent sets of order $k$ and at most $O\left((12-\nu)^{\frac{n}{4}}\right)$ maximal induced bipartite subgraphs.