For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erdös, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum number of maximal independent sets in a graph on n vertices, as well as the extremal graphs. In this paper we maximize the number of maximal independent sets of a fixed size for all graphs of order n and determine the extremal graphs. Our result generalizes the classical result.
A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine its extremal graph. The result answers a question proposed by Chakraborit and Loh. The proof uses an edge-switching operation of hypergraphs which remains the number of independent sets nondecreasing.
There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the Kruskal-Katona theorem that the lex graph has the maximum number of independent sets in a graph of fixed size and order. In this paper we solve two equivalent problems. The first is: what $3$-uniform hypergraph on a ground set of size $n$, having at least $t$ edges, has the most $2$-independent sets? Here a $2$--independent set is a subset of vertices contain…
Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $δ$. They further conjectured that when $n\geq 2δ$ and $t\geq 3$, $i_t(G)$ is maximized by the complete bipartite graph $K_{δ, n-δ}$. This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.
The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally ad…
We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number mis$(n)$ of maximal independent sets in an $n$-vertex graph was determined by Moon and Moser. The maximum number mis$_\bigtriangleup(n)$ of maximal independent sets in an $n$-vertex triangle-free graph was determined by Hujter and Tuza. We determine the maximum number mis$_t(n)$ of maximal independent sets in an $n$-vertex graph containing no induced triangle matching of size $t+1$. We also reprove a stability result of Kahn and Park on the maximum number mis$_{\bigtr…
We obtain sharp lower and upper bounds for the number of maximal (under inclusion) independent sets in trees with fixed number of vertices and diameter. All extremal trees are described up to isomorphism.
The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes …
Answering several questions of Duffus, Frankl and Rödl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the n-dimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.
We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for König-Egerváry graphs in terms of their matching numbers.
Answering questions of Y. Rabinovich, we prove "stability" versions of upper bounds on maximal independent set counts in graphs under various restrictions. Roughly these say that being close to the maximum implies existence of a large induced matching or triangle matching (depending on assumptions).
We classify the maximal $m$-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of the Johnson graph $J(n, m)$ for $m = 2, 3$. Furthermore, we determine the necessary and sufficient condition for $n$ and $m$ such that the representation of the Johnson graph $J(n, m)$ is not maximal as an $m$-distance set. Also, we classify the maximal two-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of $J(n - 1, 2)$.
Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them "stability" results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$.
Bishnoi conjectured that if a minimal t-fold blocking set in a projective plane of prime power order has maximal size then it is either a projective plane minus one point, the complement of a Baer subplane or a unital. In this note we prove this conjecture.
For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t-6, we completely characterize such graphs.
A subset of vertices $F$ in a graph $G$ is called a \emph{dissociation set} if the induced subgraph $G[F]$ of $G$ has maximum degree at most 1. A \emph{maximal dissociation set} of $G$ is a dissociation set which is not a proper subset of any other dissociation sets. A \emph{maximum dissociation set} is a dissociation set of maximum size. We show that every graph of order $n$ has at most $10^{\frac{n}{5}}$ maximal dissociation sets, and that every triangle-free graph of order $n$ has at most $6^{\frac{n}{4}}$ maximal dissociation sets. We also characterize the extremal graphs on which these up…
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $\mathbb{F}_q$. It turns out that integral point sets over $\mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point s…
We find the maximum number of maximal independent sets in two families of graphs: all graphs with $n$ vertices and at most $r$ cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs.
In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of these sets with size more than $f(k,q)=\max\{3q^4+6q^3+5q^2+q+1,θ_{k+1}+q^4+2q^3+3q^2\}$.
Counting the number of maximal independent sets of graphs was started over $50$ years ago by Erdős and Mooser. The problem has been continuously studied with a number of variations. Interestingly, when the maximal condition of an independent set is removed, such the concept presents one of topological indices in molecular graphs, the so called Merrifield-Simmons index. In this paper, we applied the concept of bivariate generating function to establish the recurrence relations of the numbers of maximal independent sets of regualr $n$-gonal cacti when $3 \leq n \leq 6$. By the ideas on meromorph…
An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by t…
We introduce a class of sets of words which is a natural common generalization of Sturmian sets and of interval exchange sets. This class of sets consists of the uniformly recurrent tree sets, where the tree sets are defined by a condition on the possible extensions of bispecial factors. We prove that this class is closed under maximal bifix decoding. The proof uses the fact that the class is also closed under decoding with respect to return words.
Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $ϕ$-open if each vertex has an incident angle of size at least $ϕ$. In this paper we study the following type of question: What is the maximum angle $ϕ$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ th…
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $\mathbb{F}_q$. It turns out that integral point sets over $\mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point s…
We determine, within 1, the value of N for which sum (s1 choose i)(s2 choose N)(s1 choose N-i)(N choose i) achieves its maximum value. Here s1 and s2 are fixed integers. This problem arises in studying the most likely value for the size of the union of A, B, and C if A and C are disjoint sets of size s1, and B is a set of size s2. Attempting to remove the 1 unit of indeterminacy leads to interesting conjectures about a family of rational functions.
Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a formula for c(n,r) when r is large relative to n, while a theorem of Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We complete the determination of c(n,r) for all n and r and characterize the extremal graphs. Problems for maximum independent sets are also completely resolved.
In 1973, Katona raised the problem of determining the maximum number of subsets in a separating cover on n elements. The answer to Katona's question turns out to be the inverse to the answer to a much simpler question: what is the largest integer which is the product of positive integers with sum n? We give a combinatorial explanation for this relationship, via Moon and Moser's answer to a question of Erdos: how many maximal independent sets can a graph on n vertices have? We conclude by showing how Moon and Moser's solution also sheds light on a problem of Mahler and Popken's about the comple…
A set $X$ in the Euclidean space $\mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $\mathbb{R}^d$ is said to be maximal if there does not exist a vector $x$ in $\mathbb{R}^d$ such that the union of $X$ and $\{x\}$ still has only $m$ distances. Bannai--Sato--Shigezumi (2012) investigated the maximal $m$-distance sets which contain the Euclidean representation of the Johnson graph $J(n,m)$. In this paper, we consider the same problem for the Hamming graph $H(n,m)$. The Euclidean representatio…
Paul Erdős and László Lovász established that any \emph{maximal intersecting family of $k-$sets} has at most $k^{k}$ blocks. They introduced the problem of finding the maximum possible number of blocks in such a family. They also showed that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. Later Péter Frankl, Katsuhiro Ota and Norihide Tokushige used a remarkable construction to prove the existence of a maximal intersecting family of $k-$sets with at least $(\frac{k}{2})^{k-1}$ blocks. In this article we introduce the notion of a \emph{closed intersec…
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sizes of sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set $D$ of positive integers, call a set $S$ "$D$-avoiding" if no two elements of $S$ differ by an element of $D$. It is shown that any $D$-avoiding set that is maximal in the class of $D$-avoiding sets (with respect to germ-ordering) is u…
Miller and Muller (1960) and independently Moon and Moser (1965) determined the maximum number of maximal independent sets in an $n$-vertex graph. We give a new and simple proof of this result.
We show that, in contrast to the integers setting, almost all even order abelian groups $G$ have exponentially fewer maximal sum-free sets than $2^{μ(G)/2}$, where $μ(G)$ denotes the size of a largest sum-free set in $G$. This confirms a conjecture of Balogh, Liu, Sharifzadeh and Treglown.
We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_ξ$, for $ξ<ω_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.
We study the following inverse graph-theoretic problem: how many vertices should a graph have given that it has a specified value of some parameter. We obtain asymptotic for the minimal number of vertices of the graph with the given number $n$ of maximal independent sets for a class of natural numbers that can be represented as concatenation of periodic binary words.
Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)
The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabeled (i.e., defined up to a rotation) max…
A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while maintaining the $m$-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only $2$ distances. We construct several $d$-dimensional maximal $2$-distance sets that contain a $d$-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical $2$-…
Given a graph $G$, viewed as a loop-less symmetric digraph, we study the maximum number of fixed points in a conjunctive boolean network with $G$ as interaction graph. We prove that if $G$ has no induced $C_4$, then this quantity equals both the number of maximal independent sets in $G$ and the maximum number of maximal independent sets among all the graphs obtained from $G$ by contracting some edges. We also prove that, in the general case, it is coNP-hard to decide if one of these equalities holds, even if $G$ has a unique induced $C_4$.
In order to better understand the structure of closed collections of reversible gates, we investigate the lattice of closed sets and the maximal members of this lattice. In this note, we find the maximal closed sets over a finite alphabet. We find that for odd sized alphabets, there are a finite number of maximal closed sets, while for the even case we have a countable infinity, almost all of which are related to an alternating permutations. We then extend to other forms of closure for reversible gates, ancilla and borrow closure. Here we find some structural results, including some examples o…
We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.
Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and every additional point $p_{n+1}$ destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove …
We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.
Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our m…
We show that, for any n-vertex graph G and integer parameter k, there are at most 3^{4k-n}4^{n-3k} maximal independent sets I \subset G with |I| <= k, and that all such sets can be listed in time O(3^{4k-n} 4^{n-3k}). These bounds are tight when n/4 <= k <= n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 3^{4/3}/4)^n) ~= 2.4150^n, improving a previous O((1+3^{1/3})^n) ~= 2.4422^n algorithm of Lawler (1976).
We investigate the existence of maximal collections of mutually noncrossing $k$-element subsets of $\left\{ 1, \dots, n \right\}$ that are invariant under adding $k\pmod n$ to all indices. Our main result is that such a collection exists if and only if $k$ is congruent to $0, 1$ or $-1$ modulo $n/\operatorname{GCD}(k,n)$. Moreover, we present some algebraic consequences of our result related to self-injective Jacobian algebras.
