Independent Set Of Size Research Articles

Degree Ramsey numbers for cycles and blowups of trees

Let H→sG mean that every s-coloring of E(H) produces a monochromatic copy of G in some color class. Let the s-color degree Ramsey number of a graph G, written RΔ(G;s), be min{Δ(H):H→sG}. We prove that the 2-color degree Ramsey number is at most 96 for every even cycle and at most 3458 for every odd cycle. For the general s-color problem on even cycles, we prove RΔ(C2m;s)≤16s6 for all m, and RΔ(C4;s)≥0.007s14/9. The constant upper bound forRΔ(Cn;2) uses blowups of graphs, where the d-blowup of a graph G is the graph G′ obtained by replacing each vertex of G with an independent set of size d and each edge e of G with a copy of the complete bipartite graph Kd,d. We also prove the existence of a function f such that if G′ is the d-blowup of G, then RΔ(G′;s)≤f(RΔ(G;s),s,d).

Upper Bounds for Erdös-Hajnal Coefficients of Tournaments

A celebrated unresolved conjecture of Erdös and Hajnal (see Discrete Appl Math 25 (1989), 37–52) states that for every undirected graph H, there exists , such that every graph on n vertices which does not contain H as an induced subgraph contains either a clique or an independent set of size at least . In (Combinatorica (2001), 155–170), Alon et al. proved that this conjecture was equivalent to a similar conjecture about tournaments. In the directed version of the conjecture cliques and stable sets are replaced by transitive subtournaments. For a fixed undirected graph H, define to be the supremum of all ε for which the following holds: for some n0 and every every undirected graph with vertices not containing H as an induced subgraph has a clique or independent set of size at least . The analogous definition holds if H is a tournament. We call the Erdös–Hajnal coefficient of H. The Erdös–Hajnal conjecture is true if and only if for every H. We prove in this article that: the Erdös–Hajnal coefficient of every graph H is at most , there exists such that the Erdös–Hajnal coefficient of almost every tournament T on k vertices is at most , i.e. the proportion of tournaments on k vertices with the coefficient exceeding goes to 0 as k goes to infinity.

Elimination graphs

In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = ( V, E ) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v 1 , …, v n such that the property P holds on the induced subgraph G [ N ( v i )∩ V i ], where N ( v i ) is the neighborhood of v i and V i = { v i , …, v n }. It turns out that if we take P as a graph with maximum independent set size no greater than k , then this definition gives a natural generalization of both chordal graphs and ( k + 1)-claw-free graphs. We refer to such graphs as inductive k -independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k -independent for small k . In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3 -independent graph; furthermore, any planar graph is an inductive 3 -independent graph. For any fixed constant k , we develop simple, polynomial time approximation algorithms for inductive k -independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.

Counting Subgraphs via Homomorphisms

We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well-known results in algorithms and combinatorics, including the recent algorithm of Björklund, Husfeldt, and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn et al. for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color-coding-based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions, and the color coding technique, we obtain the following new results. The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time $ 2^{n+o(n)} $, in particular, in time $\mathcal{O}(2^{n}n^3)$ for trees and in time $2^{n+\mathcal{O}(\sqrt{n})}$ for planar graphs. Counting all maximum planar subgraphs, subgraphs of bounded genus, or more generally subgraphs excluding a fixed graph M as a minor can be done in $2^{\mathcal{O}(n)}$ time. Counting all subtrees with a given maximum degree (a generalization of counting Hamiltonian paths) of a given graph can be done in time $2^{\mathcal{O}(n)}$. A generalization of Ryser's formula is, Let G be a graph with an independent set of size $\ell$. Then the number of perfect matchings in G can be found in time $\mathcal{O}(2^{n-\ell} n^3)$. Let ${\cal H}$ be a graph class excluding a fixed graph M as a minor. Then the maximum number of vertex disjoint subgraphs from ${\cal H}$ in a graph G on n vertices can be found in time $2^{\mathcal{O}(n)}$. In order to show this, we prove that there exists a constant $c_M$ depending only on M such that the number of nonisomorphic n-vertex graphs in ${\cal H}$ is at most $c_M^n$. Let F be a k-vertex graph of treewidth t and let G be an n-vertex graph. A subgraph of G isomorphic to F (if one exists) can be found in $\mathcal{O}(4.32^k \cdot k \cdot t \cdot n^{t+1})$ expected time using $\mathcal{O}(\log{k} \cdot n^{t+1})$ space.

Modular geometry of the symplectic group attached to a q-level system and to multiple q-dit mixtures

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q = 4) and two-qubit (q = 22) systems, the octit (q = 8), qubit/quartit (q = 2×4) and three-qubit (q = 23) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12,…, one defines a bijection between the σ(q) maximal commuting sets [with σ[q)the sum of divisors of q] of Pauli observables and the maximal submodules of the modular ring ℤq2, that arrange into the projective line P1(ℤq) and a independent set of size σ(q) − ψ(q) [with ψ(q) the Dedekind psi function]. In the multiple qudit case, e.g. q = 22, 23, 32,…, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if q = 22) and GQ(3,3) (if q = 32). More precisely, in dimension pn (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field Fp. In this space, one makes use of the commutator to define a symplectic polar space W2n−1(p) of cardinality σ(p2n−1), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n−1(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p2N−1). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.

Pauli graphs when the Hilbert space dimension contains a square: Why the Dedekind psi function?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low-dimensional examples are the quartit (q = 4) and two-qubit (q = 22) systems, the octit (q = 8), qubit/quartit (q = 2 × 4) and three-qubit (q = 23) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12, … , one defines a bijection between the σ(q) maximal commuting sets (with σ[q) being the sum of divisors of q) of Pauli observables and the maximal submodules of the modular ring , that arrange into the projective line and an independent set of size σ(q) − ψ(q) (with ψ(q) being the Dedekind psi function). In the multiple qudit case, e.g. q = 22, 23, 32, … , the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2, 2) (if q = 22) and GQ(3, 3) (if q = 32). More precisely, in the dimension pn (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field . In this space, one makes use of the commutator to define a symplectic polar space W2n − 1(p) of cardinality σ(p2n − 1) that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n − 1(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p2n − 1). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, punctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.

Fractional colorings of cubic graphs with large girth

We show that every (sub)cubic -vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978, which implies that it contains an independent set of size at least . Our bound on the independence number is valid for random cubic graphs as well, as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.

Automorphism groups of root system matroids

Given a root system R , the vector system R ̃ is obtained by taking a representative v in each antipodal pair { v , − v } . The matroid M ( R ) is formed by all independent subsets of R ̃ . The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root system matroids M ( R ) are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root system matroids.

A note on propositional proof complexity of some Ramsey-type statements

A Ramsey statement denoted \({n \longrightarrow (k)^2_2}\) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n k) and with terms of size \({\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}\) . Let r k be the minimal n for which the statement holds. We prove that RAM(r k , k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4k, k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.

Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

The Ramsey number r(H, K n ) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Turan number ex(m, H) is the maximu...

A note on regular Ramsey graphs

AbstractWe prove that there is an absolute constant C>0 so that for every natural n there exists a triangle‐free regular graph with no independent set of size at least \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}\({{C}}\sqrt{{{n}}\log{{n}}}\).\end{document} © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 244–249, 2010

Dilworth's Decomposition Theorem for Posets

Dilworth's Decomposition Theorem for PosetsThe following theorem is due to Dilworth [8]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques).In this article we formalize an elegant proof of the above theorem for finite posets by Perles [13]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [8].A dual of Dilworth's theorem also holds: a poset with maximum clique m is a union of m independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [11]. Mirsky states also a corollary that a poset of r x s + 1 elements possesses a clique of size r + 1 or an independent set of size s + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [9].Instead of using posets, we drop reflexivity and state the facts about anti-symmetric and transitive relations.

Delaunay graphs of point sets in the plane with respect to axis‐parallel rectangles

AbstractGiven a point set P in the plane, the Delaunay graph with respect to axis‐parallel rectangles is a graph defined on the vertex set P, whose two points p,q ∈ P are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94–136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(nlog 2 log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn log log n/(log n log log log n).We give two further applications of our methods: (1) We construct two‐dimensional n‐element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473–475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41–48). (2) For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis‐parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer‐Verlag, New York, 2005). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O ∗ ( 2 n − z ) , where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O ∗ ( 2 n / 2 ) . Another implication is an algorithm for general graphs whose running time is O ( 1.7088 n ) .

The Resolution Complexity of Independent Sets and Vertex Covers in Random Graphs

We consider the problem of providing a resolution proof of the statement that a given graph with n vertices and average degree roughly Δ does not contain an independent set of size k. For randomly chosen graphs and k � n/3, we show that such proofs asymptotically almost surely require size roughly exponential in n/Δ6. This, in particular, implies a 2� (n) lower bound for constant degree graphs, and for Δ � n 1/6, shows that there are almost always no short resolution proofs for k as large as n/3 even though a maximum independent set is likely to be much smaller, roughly n 5/6 in size. Our result implies that for graphs that are not too dense, almost all instances of the independent set problem are hard for resolution. Further, it provides an unconditional exponential lower bound on the running time of resolution-based search algorithms for finding a maximum independent set or approximating it within a factor of Δ/(6 ln Δ). We also give relatively simple upper bounds for the problem and show them to be tight for the class of exhaustive backtracking algorithms. We deduce similar complexity results for the related vertex cover problem on random graphs, proving, in particular, that no polynomial-time resolution-based method can achieve an approximation within a factor of 3/2.

On graphs with subgraphs having large independence numbers

AbstractLet G be a graph on n vertices in which every induced subgraph on ${s}={\log}^{3}\, {n}$ vertices has an independent set of size at least ${t}={\log}\, {n}$. What is the largest ${q}={q}{(n)}$ so that every such G must contain an independent set of size at least q? This is one of the several related questions raised by Erdős and Hajnal. We show that ${q}{(n)}= \Theta({\log}^{2} {n}/{\log}\, {\log} \,{n})$, investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey‐type problem. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 149–157, 2007

A Note on the Use of Independent Sets for the k-SAT Problem

An independent set of variables is one in which no two variables occur in the same clause in a given k-SAT instance. Recently, independent sets have obtained more attention. Due to a simple observation we prove that a k-SAT instance over n variables with independent set of size i can be solved in time O(φ_{2(k−1)} (n − i)) where φ_{k (n)} denotes an upper bound on the complexity of solving k-SAT over n variables

Finding Large Independent Sets in Polynomial Expected Time

We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. Let $G_{n,p}$ be a random graph, and let $S$ be a set of $k$ vertices, chosen uniformly at random. Then, let $G_0$ be the graph obtained by deleting all edges connecting two vertices in $S$. Finally, an adversary may add edges to $G_0$ that do not connect two vertices in $S$, thereby producing the instance $G=G_{n,p,k}^*$. We present an algorithm that on input $G=G_{n,p,k}^*$ finds an independent set of size $\geq k$ within polynomial expected time, provided that $k\geq C(n/p)^{1/2}$ for a certain constant $C>0$. Moreover, we prove that in the case $k\leq (1-\varepsilon)\ln(n)/p$ this problem is hard.

Infinite independent sets in distributive lattices

We show that a poset P contains a subset isomorphic to \([\kappa ]^{ < \omega }\)if and only if the poset J(P) consisting of ideals of P contains a subset isomorphic to \(\mathcal{P}(\kappa ),\) the power set of κ. If P is a join-semilattice this amounts to the fact that P contains an independent set of size κ. We show that if κ := ω and P is a distributive lattice, then this amounts to the fact that P contains either \(I_{ < \omega } (\Gamma )\) or \(I_{ < \omega } (\Delta )\) as sublattices, where Γ and Δ are two special meet-semilattices already considered by J. D. Lawson, M. Mislove and H. A. Priestley.

Bounded-Degree Independent Sets in Planar Graphs

An independent set of a graph is a set of vertices without edges between them. Every planar graph has an independent set of size at least (1/4)n and there are planar graphs for which any independent set has size at most (1/4)n. In this paper similar bounds are provided for the problem of bounded-degree independent set, i.e., an independent set where additionally all vertices have degree less than a pre-specified bound D. Our upper and lower bounds match (up to a small constant) for D ? 16.