The maximum number of maximal independent sets in unicyclic connected graphs

In this paper, we determine the maximum number of maximal independent sets in a unicyclic connected graph. We also find a class of graphs achieving this maximum value.     

On independent sets in hypergraphs

The independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove that if H_n is an n‐vertex ‐uniform hypergraph in which every r‐element set is contained in at most d edges, where , then where satisfies as . The value of c_r improves and generalizes several earlier results that all use a theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi (J Comb Theory Ser A 32 (1982), 321–335). Our relatively short proof extends a method due to Shearer (Random Struct Algorithms 7 (1995), 269–271) and Alon (Random Struct Algorithms 9 (1996), 271–278). The above statement is close to best possible, in the sense that for each and all values of , there are infinitely many H_n such that where depends only on r. In addition, for many values of d we show as , so the result is almost sharp for large r. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 224‐239, 2014     

Rapid mixing of hypergraph independent sets

We prove that the mixing time of the Glauber dynamics for sampling independent sets on n‐vertex k‐uniform hypergraphs is when the maximum degree Δ satisfies Δ ≤ c2^k/2, improving on the previous bound Bordewich and co‐workers of Δ ≤ k ? 2. This result brings the algorithmic bound to within a constant factor of the hardness bound of Bezakova and co‐workers which showed that it is NP‐hard to approximately count independent sets on hypergraphs when Δ ≥ 5·2^k/2.     

Projectivity and independent sets in powers of graphs

We investigate the relationship between projectivity and the structure of maximal independent sets in powers of circular graphs, Kneser graphs and truncated simplices. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 162–171, 2002     

Estimates of the number of independent sets in graphs with a fixed independence number

Attainable estimates of the number of independent sets in graphs with a given size of the maximal independent set are obtained. Three graph classes—trees, forests, and the class of all graphs—are considered. Extremal graphs are described.     

The structure and maximum number of maximum independent sets in trees

A subset of vertices is a maximum independent set if no two of the vertices are joined by an edge and the subset has maximum cardinality. in this paper we answer a question posed by Herb Wilf. We show that the greatest number of maximum independent sets for a tree of n vertices is We give the families of trees on which these maxima are achieved. Proving which trees are extremal depends upon the structure of maximum independent sets in trees. This structure is described in terms of adjacency rules between three types of vertices, those which are in all, no, or some maximum independent sets. We show that vertices that are in some but not all maximum independent sets of the tree are joined in pairs by the α-critical edges (edges whose removal increases the size of a maximum independent set). The number of maximum independent sets is shown to depend on the structure within the tree of the α-critical edges.     

Minimizing the number of independent sets in triangle-free regular graphs

Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin–Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give a new bound on the number of independent sets in triangle-free cubic graphs.     

Using critical sets to solve the maximum independent set problem

A method that utilizes the polynomially solvable critical independent set problem for solving the maximum independent set problem on graphs with a nonempty critical independent set is developed. The effectiveness of the proposed approach on large graphs with large independence number is demonstrated through extensive numerical experiments.     

Graphs with the second largest number of maximal independent sets

Let G be a simple undirected graph. Denote by (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. Erd?s and Moser raised the problem of determining the maximum value of among all graphs of order n and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most r cycles. In this paper we determine the second largest value of and xi(G) among all graphs of order n. Moreover, the extremal graphs achieving these values are also determined.     

Trees with the second largest number of maximal independent sets

A maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we determine the second largest number of maximal independent sets among all trees and forests of order n≥4. We also characterize those extremal graphs achieving these values.     

On the maximum number of minimum dominating sets in forests

Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $\gamma$ has at most $2^{\gamma }$ minimum dominating sets. Bień gave a counterexample, which allows us to construct forests with domination number $\gamma$ and $2.0598^{\gamma }$ minimum dominating sets. We show that every forest with domination number $\gamma$ has at most $2.4606^{\gamma }$ minimum dominating sets, and that every tree with independence number $\alpha$ has at most $2^{\alpha -1}+1$ maximum independent sets.     

Probabilistic analysis of a parallel algorithm for finding maximal independent sets

We consider a natural parallel version of the classical greedy algorithm for finding a maximal independent set in a graph. This version was studied in Coppersmith, Raghavan, and Tompa and they conjecture there that its expected running time on random graphs of arbitrary edge density of O (log n). We prove that conjecture.     

The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

A set A of non‐negative integers is called a Sidon set if all the sums , with and a₁, , are distinct. A well‐known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erd?s, Singer and Turán from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non‐differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erd?s (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015     

Maximizing the number of independent subsets over trees with bounded degree

The number of independent vertex subsets is a graph parameter that is, apart from its purely mathematical importance, of interest in mathematical chemistry. In particular, the problem of maximizing or minimizing the number of independent vertex subsets within a given class of graphs has already been investigated by many authors. In view of the applications of this graph parameter, trees of restricted degree are of particular interest. In the current article, we give a characterization of the trees with given maximum degree which maximize the number of independent subsets, and show that these trees also minimize the number of independent edge subsets. The structure of these trees is quite interesting and unexpected: it can be described by means of a novel digital system—in the case of maximum degree 3, we obtain a binary system using the digits 1 and 4. The proof mainly depends on an exchange lemma for branches of a tree. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 49–68, 2008     

Maximal and maximum independent sets in graphs with at most r cycles

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. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 283–314, 2006     

Maximal independent sets in caterpillar graphs

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (intersection graph of mis) and the clique graph (intersection graph of cliques) of complete caterpillar graphs.     

Uniqueness of maximum planar five-distance sets

A subset X in the Euclidean plane is called a k-distance set if there are exactly k distances between two distinct points in X. We denote the largest possible cardinality of k-distance sets by g(k). Erdős and Fishburn proved that g(5)=12 and also conjectured that 12-point five-distance sets are unique up to similar transformations. We classify 8-point four-distance sets and prove the uniqueness of the 12-point five-distance sets given in their paper. We also introduce diameter graphs of planar sets and characterize these graphs.     

Another note on cliques and independent sets

Entringer, Goddard, and Henning studied graphs in which every vertex belongs to both an (m + 1)‐clique and an independent (n + 1)‐set; they proved that there is such a graph of order p if and only if . We give an alternative and slightly easier proof of this fact, relating it to combinatorial matrix theory. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 173–175, 2000     

偶错位图的张量幂的最大独立集和自同构群

若A_n 是X := {1, 2,..., n} 上的偶置换构成的交错群, E_n 是X 上的偶错位集, 则Cayley 图AΓ_n := Γ(A_n, E_n) 称为偶错位图. 令AΓ_n^q 为q 个AΓ_n 的张量幂. 在本文中, 我们研究了AΓ_n^q 的连通性、直径、独立数、团数、色数和最大独立集等性质. 利用AΓ_n^q 最大独立集的结果, 我们完全确定了AΓ_n^q 的自同构群的结构.