22 research outputs found
On the number of maximal independent sets in a graph
Miller and Muller (1960) and independently Moon and Moser (1965) determined
the maximum number of maximal independent sets in an -vertex graph. We give
a new and simple proof of this result
On the number of -transversals in hypergraphs
A set of vertices in a hypergraph is \textit{strongly independent} if
every hyperedge shares at most one vertex with . We prove a sharp result for
the number of maximal strongly independent sets in a -uniform hypergraph
analogous to the Moon-Moser theorem.
Given an -uniform hypergraph and a non-empty set of
non-negative integers, we say that a set is an \textit{-transversal} of
if for any hyperedge of , we have
\mbox{}. Independent sets are
-transversals, while strongly independent sets are
-transversals. Note that for some sets , there may exist
hypergraphs without any -transversals. We study the maximum number of
-transversals for every , but we focus on the more natural sets, e.g.,
, or being the set of odd or the set of even
numbers.Comment: 10 page
On the phase transitions of graph coloring and independent sets
We study combinatorial indicators related to the characteristic phase
transitions associated with coloring a graph optimally and finding a maximum
independent set. In particular, we investigate the role of the acyclic
orientations of the graph in the hardness of finding the graph's chromatic
number and independence number. We provide empirical evidence that, along a
sequence of increasingly denser random graphs, the fraction of acyclic
orientations that are `shortest' peaks when the chromatic number increases, and
that such maxima tend to coincide with locally easiest instances of the
problem. Similar evidence is provided concerning the `widest' acyclic
orientations and the independence number
On the Number of Maximal Bipartite Subgraphs of a Graph
We show new lower and upper bounds on the number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105^{n/10} ~= 1.5926^n such subgraphs, which improves an earlier lower bound by Schiermeyer (1996). We show an upper bound of n . 12^{n/4} ~= n . 1.8613^n and give an algorithm that lists all maximal induced bipartite subgraphs in time proportional to this bound. This is used in an algorithm for checking 4-colourability of a graph running within the same time bound
Sharp bound on the number of maximal sum-free subsets of integers
Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free
sets in is much smaller than the number of sum-free sets. In
the same paper they gave a lower bound of for the
number of maximal sum-free sets. Here, we prove the following: For each , there is a constant such that, given any ,
contains maximal sum-free sets. Our
proof makes use of container and removal lemmas of Green, a structural result
of Deshouillers, Freiman, S\'os and Temkin and a recent bound on the number of
subsets of integers with small sumset by Green and Morris. We also discuss
related results and open problems on the number of maximal sum-free subsets of
abelian groups.Comment: 25 pages, to appear in the Journal of the European Mathematical
Societ
The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable
Monadic second order logic can be used to express many classical notions of
sets of vertices of a graph as for instance: dominating sets, induced
matchings, perfect codes, independent sets or irredundant sets. Bounds on the
number of sets of any such family of sets are interesting from a combinatorial
point of view and have algorithmic applications. Many such bounds on different
families of sets over different classes of graphs are already provided in the
literature. In particular, Rote recently showed that the number of minimal
dominating sets in trees of order is at most and that
this bound is asymptotically sharp up to a multiplicative constant. We build on
his work to show that what he did for minimal dominating sets can be done for
any family of sets definable by a monadic second order formula.
We first show that, for any monadic second order formula over graphs that
characterizes a given kind of subset of its vertices, the maximal number of
such sets in a tree can be expressed as the \textit{growth rate of a bilinear
system}. This mostly relies on well known links between monadic second order
logic over trees and tree automata and basic tree automata manipulations. Then
we show that this "growth rate" of a bilinear system can be approximated from
above.We then use our implementation of this result to provide bounds on the
number of independent dominating sets, total perfect dominating sets, induced
matchings, maximal induced matchings, minimal perfect dominating sets, perfect
codes and maximal irredundant sets on trees. We also solve a question from D.
Y. Kang et al. regarding -matchings and improve a bound from G\'orska and
Skupie\'n on the number of maximal matchings on trees. Remark that this
approach is easily generalizable to graphs of bounded tree width or clique
width (or any similar class of graphs where tree automata are meaningful)
The minimum number of maximal independent sets in twin-free graphs
The problem of determining the maximum number of maximal independent sets in
certain graph classes dates back to a paper of Miller and Muller and a question
of Erd\H{o}s and Moser from the 1960s. The minimum was always considered to be
less interesting due to simple examples such as stars. In this paper we show
that the problem becomes interesting when restricted to twin-free graphs, where
no two vertices have the same open neighbourhood. We consider the question for
arbitrary graphs, bipartite graphs and trees. The minimum number of maximal
independent sets turns out to be logarithmic in the number of vertices for
arbitrary graphs, linear for bipartite graphs and exponential for trees. In the
latter case, the minimum and the extremal graphs have been determined earlier
by Taletski\u{\i} and Malyshev, but we present a shorter proof.Comment: 17 pages, 7 figures, 5 table