The typical structure of maximal triangle-free graphs

Recently, settling a question of Erd\H{o}s, Balogh and Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest.Comment: 17 page

On the number of k-dominating independent sets

We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of $k$-dominating independent sets in $n$-vertex graphs is between $c_k\cdot\sqrt[2k]{2}^n$ and $c_k'\cdot\sqrt[k+1]{2}^n$ if $k\geq 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$ and $c'\cdot1.246^n$. Graph constructions containing a large number of $k$-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.Comment: 13 page

Maximal independent sets in graphs with at most one cycle

AbstractIn this paper, we determine the largest number of maximal independent sets among all connected graphs of order n, which contain at most one cycle. We also characterize those extremal graphs achieving this maximum value. As a consequence, the corresponding results for graphs with at most one cycle but not necessarily connected are also given