Number Of Independent Sets Research Articles

The number of independent sets in bipartite graphs and benzenoids

The number of independent sets in bipartite graphs and benzenoids

New Ramsey Multiplicity Bounds and Search Heuristics

AbstractWe study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erdős. Most notably, we improve the upper bounds on the Ramsey multiplicity of $$K_4$$ K 4 and $$K_5$$ K 5 and settle the minimum number of independent sets of size 4 in graphs with clique number at most 4. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic $$K_4$$ K 4 or $$K_5$$ K 5 in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a graph of constant size and were found through search heuristics. They are complemented by lower bounds established using flag algebras, resulting in a fully computer-assisted approach. For some of our theorems we can also derive that the extremal construction is stable in a very strong sense. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs.

Extremal graphs for the number of independent sets on polygonal chains joined by cutting edges

Extremal graphs for the number of independent sets on polygonal chains joined by cutting edges

Independence polynomials of graphs with given cover number or dominate number

For a graph [Formula: see text], let [Formula: see text] be the number of independent sets of size [Formula: see text] in [Formula: see text]. The independence polynomial [Formula: see text] has been the focus of considerable research. In this paper, using the coefficients of independence polynomials, we order graphs with some given parameters. We first determine the extremal graph whose all coefficients of [Formula: see text] are the largest (respectively, smallest) among all connected graphs (respectively, bipartite graphs) with given vertex cover number. Then we also derive the extremal graph whose all coefficients of [Formula: see text] are the largest among all connected graphs (respectively, bipartite graphs) with given vertex dominate number.

Log-concave poset inequalities

We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concave inequal- ities for counting certain weighted feasible words, which generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. Notably, we prove matching equality conditions for both earlier inequalities and our extensions. In contrast with much of the previous work, our proofs are combinatorial and employ nothing but linear algebra. We use the language formulation of greedoids which allows a linear algebraic setup, which in turn can be analyzed recursively. The underlying non- commutative nature of matrices associated with greedoids allows us to proceed beyond polymatroids and prove the equality conditions. As further application of our tools, we rederive both Stanley’s inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.

Counting Rules for Computing the Number of Independent Sets of a Grid Graph

The issue of counting independent sets of a graph, G, represented as i(G), is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry, i(G) is recognized as the Merrifield–Simmons (M-S) index for molecular graphs, which is one of the most relevant topological indices related to the boiling point in chemical compounds. This article introduces an innovative algorithm designed for tallying independent sets within grid-like structures. The proposed algorithm is based on the ‘branch-and-bound’ technique and is applied to compute i(Gm,n) for a square grid formed by m rows and n columns. The proposed approach incorporates the widely recognized vertex reduction rule as the basis for splitting the current subgraph. The methodology involves breaking down the initial grid iteratively until outerplanar graphs are achieved, serving as the ’basic cases’ linked to the leaf nodes of the computation tree or when no neighborhood is incident to a minimum of five rectangular internal faces. The time complexity of the branch-and-bound algorithm speeds up the computation of i(Gm,n) compared to traditional methods, like the transfer matrix method. Furthermore, the scope of the proposed algorithm is more general than the algorithms focused on grids since it could be applied to process general mesh graphs.

Log-concave polynomials III: Mason’s ultra-log-concavity conjecture for independent sets of matroids

We give a self-contained proof of the strongest version of Mason’s conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

Inapproximability of counting independent sets in linear hypergraphs

It is shown in this note that approximating the number of independent sets in a k-uniform linear hypergraph with maximum degree at most Δ is NP-hard if Δ≥5⋅2k−1+1. This confirms that for the relevant sampling and approximate counting problems, the regimes on the maximum degree where the state-of-the-art algorithms work are tight, up to some small factors. These algorithms include: the approximate sampler and randomised approximation scheme by Hermon et al. (2019) [5], the perfect sampler by Qiu et al. (2022) [6], and the deterministic approximation scheme by Feng et al. (2023) [7].

The total Betti number of the independence complex of ternary graphs

Given a graph G , the independence complex I(G) is the simplicial complex whose faces are the independent sets of V(G) . Let \tilde{b}_i denote the i -th reduced Betti number of I(G) , and let b(G) denote the sum of the \tilde{b}_i(G) ’s. A graph is ternary if it does not contain induced cycles with length divisible by 3. Kalai and Meshulam conjectured that b(G)\le 1 whenever G is ternary. We prove this conjecture. This extends a recent result proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph G , the number of independent sets with even cardinality and the number of independent sets with odd cardinality differ by at most 1.

Homomorphisms from the torus

We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus Zmn, where m is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper q-colourings of Zmn (so in particular, the discrete hypercube). We give further applications to the study of height functions and (generalised) rank functions on the discrete hypercube and disprove a conjecture of Kahn and Lawrenz. For the proof we combine methods from statistical physics, entropy and graph containers and exploit isoperimetric and algebraic properties of the torus.

Counting colorings of triangle-free graphs

By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has chromatic number at most (C+o(1))Δ/log⁡Δ for some universal constant C>0. Using the entropy compression method, Molloy proved that one can in fact take C=1. Here we show that for every q⩾(1+o(1))Δ/log⁡Δ, the number c(G,q) of proper q-colorings of G satisfiesc(G,q)⩾(1−1q)m((1−o(1))q)n, where n=|V(G)| and m=|E(G)|. Except for the o(1) term, this lower bound is best possible as witnessed by random Δ-regular graphs. When q=(1+o(1))Δ/log⁡Δ, our result yields the inequalityc(G,q)⩾exp⁡((1−o(1))log⁡Δ2n), which improves an earlier bound of Iliopoulos and yields the optimal value for the constant factor in the exponent. Furthermore, this result implies the optimal lower bound on the number of independent sets in G due to Davies, Jenssen, Perkins, and Roberts. An important ingredient in our proof is the counting method that was recently developed by Rosenfeld. As a byproduct, we obtain an alternative proof of Molloy's bound χ(G)⩽(1+o(1))Δ/log⁡Δ using Rosenfeld's method in place of entropy compression (other proofs of Molloy's theorem using Rosenfeld's technique were given independently by Hurley and Pirot and Martinsson).

Approximately counting independent sets in bipartite graphs via graph containers

AbstractBy implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to ‐regular, bipartite graphs satisfying a weak expansion condition: when is constant, and the graph is a bipartite ‐expander, we obtain an FPTAS for the number of independent sets. Previously such a result for was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a ‐regular, bipartite ‐expander, with fixed, we give an FPTAS for the hard‐core model partition function at fugacity . Finally we present an algorithm that applies to all ‐regular, bipartite graphs, runs in time , and outputs a ‐approximation to the number of independent sets.

Improved Approximation Algorithms for Bin Packing with Conflicts

Given a set of items, and a conflict graph defined on the item set, the problem of bin packing with conflicts asks for a partition of items into a minimum number of independent sets so that the total size of items in each independent set does not exceed the bin capacity. As a generalization of both classic bin packing and classic vertex coloring, it is hard to approximate the problem on general graphs. We present new approximation algorithms for bipartite graphs and split graphs. The absolute approximation ratios are shown to be [Formula: see text] and [Formula: see text] respectively, both improving the existing results.

Maximizing the number of independent sets in claw-free cubic graphs

Let G be a graph. An independent set of G is a subset of vertices no two of which are connected by an edge. Denote by I(G) the set of all independent sets of G. The independence polynomial of G is I(G;λ)=∑I∈I(G)λ|I|.The cartesian product C3□K2 is commonly called the triangular-prism, it is also a generalized Petersen graph on 6 vertices and denoted by GP(3,1). In this paper, we use the occupancy method to show that for any claw-free cubic graph G,|I(G)|≤|I(GP(3,1))||V(G)|/6,furthermore, for any λ∈(0,1541],I(G;λ)≤(I(GP(3,1);λ)|V(G)|/6,with equality holding in both cases if and only if G is a disjoint union of copies of GP(3,1).

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions.

Faster Graph Coloring in Polynomial Space

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

A Method for Computing the Merrifield–Simmons Index on Benzenoid Systems

We present a method for computing the Merrifield-Simmons index on some basic graphs. For example, our proposal works on paths, simple cycles, trees, and we show that the method can be extended to process benzenoid systems Hr,t with a total of r.t hexagons. In the case of benzenoid systems, our method consists of performing a linear-time Hamiltonian walking on an isomorphic hexagonal grid from , at the same time that the number of independent sets are being computed incrementally. Our method improves the asymptotic behavior of the time-complexity with respect to the transfer matrix method, which is the classic method for computing the Merrifield-Simmons index on grid graphs, even for benzenoid systems.

An FPTAS for the hardcore model on random regular bipartite graphs

We give a fully polynomial-time approximation scheme (FPTAS) to compute the partition function of the hardcore model of fugacity λ on random Δ-regular bipartite graphs for all sufficiently large Δ and λ≥4(log⁡Δ)3/Δ. For the special case of λ=1, where the partition function computes the number of independent sets, an FPTAS exists for Δ≥50. Our technique is based on the polymer model, which is used by Jenssen, Keevash and Perkins (SODA, 2019) to obtain an FPTAS for #BIS-hard problems for the first time. The technique also applies to counting q-colorings: For q≥3 and Δ≥Δ(q), there is an FPTAS to compute the number of q-colorings on random Δ-regular bipartite graphs.

The Expected Value of Hosoya Index and Merrifield–Simmons Index in a Random Cyclooctylene Chain

The Hosoya index m(G) and the Merrifield–Simmons index i(G) of a graph G are the number of matchings and the number of independent sets in G. In this paper, we establish exact formulas for the expected value of the Hosoya index and Merrifield–Simmons index of the random cyclooctylene chains, which are graphs of a chemical chain consisting of n octagons, each of which is connected to the end of the previous octagon by an edge. In addition, we obtain the expected values and the average values of the two indexes through the relevant chemical diagrams and a series of accurate formulas with respect to the set of all cyclooctylene chains with n octagons.

The homotopy type of the independence complex of graphs with no induced cycles of length divisible by 3

We prove Engström’s conjecture that the independence complex of graphs with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. Our result strengthens a result by Zhang and Wu, verifying a conjecture of Kalai and Meshulam which states that the total Betti number of the independence complex of such a graph is at most 1. A weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spirkl, who showed that in such a graph, the number of independent sets of even size minus the number of independent sets of odd size has values 0, 1, or −1.