19,531 research outputs found
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
-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 admits a
vertex partition such that is a perfect matching and 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 's, which is of independent interest.Comment: 17 page
Maximal induced matchings in triangle-free graphs
An induced matching in a graph is a set of edges whose endpoints induce a
-regular subgraph. It is known that any -vertex graph has at most
maximal induced matchings, and this bound is best
possible. We prove that any -vertex triangle-free graph has at most maximal induced matchings, and this bound is attained by any
disjoint union of copies of the complete bipartite graph . Our result
implies that all maximal induced matchings in an -vertex triangle-free graph
can be listed in time , yielding the fastest known algorithm for
finding a maximum induced matching in a triangle-free graph.Comment: 17 page
On the number of k-dominating independent sets
We study the existence and the number of -dominating independent sets in
certain graph families. While the case 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 -dominating
independent sets in -vertex graphs is between and
if , moreover the maximum number of
-dominating independent sets in -vertex graphs is between
and . Graph constructions containing a large number of
-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
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