Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.     

Cliques in Steiner systems

A partial Steiner (k,l)-system is a k-uniform hypergraph with the property that every l-element subset of V is contained in at most one edge of . In this paper we show that for given k,l and t there exists a partial Steiner (k,l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of this note, we establish asymptotic lower and upper bounds on the size of such cliques with respect to the order of Steiner systems. Research of the second author partially supported by NSERC grant OGP0025112.     

Distance Graphs with Large Chromatic Number and without Cliques of Given Size in the Rational Space

We study distance graphs with exponentially large chromatic number which do not contain cliques of prescribed size in the rational space.

     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

Contractible Cliques in k-Connected Graphs

Kawarabayashi proved that for any integer k≥4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge, or admits a k-contractible triangle. This implies Thomassen's result that every triangle-free k-connected graph contains a k-contractible edge. In this paper, we extend Kawarabayashi's technique and prove a more general result concerning k-contractible cliques. Xingxing Yu was partially supported by NSF grant DMS-0245530 and NSA grant MDA-904-03-1-0052.     

Triangulated Spheres and Colored Cliques

   Abstract. In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S ^d-1 can be extended to a triangulation of the ball B ^d , by adding one point at a time, having degree at most 2d to its predecessors.     

Cliques in dense inhomogeneous random graphs

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant of the Erd?s–Rényi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of for each fixed n , and give examples of graphons for which exhibits wild long‐term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably‐partite. © The Authors Random Structures & Algorithms Published byWiley Periodicals, Inc. Random Struct. Alg., 2016 © 2017 The Authors Random Structures & Algorithms Published by Wiley Periodicals, Inc. Random Struct. Alg., 51, 275–314, 2017     

Cliques that are tolerance digraphs

In this paper, we study directed graph versions of tolerance graphs, in particular, the class of totally bounded bitolerance digraphs and several subclasses. When the underlying graph is complete, we prove that the classes of totally bounded bitolerance digraphs and interval catch digraphs are equal, and this implies a polynomial-time recognition algorithm for the former class. In addition, we give examples (whose underlying graphs are complete) to separate every other pair of subclasses, and one of these provides a counterexample to a conjecture of Maehara (1984).     

Cliques, minors and apex graphs

In this paper, we proved the following result: Let G be a (k+2)-connected, non-(k−3)-apex graph where k≥2. If G contains three k-cliques, say L₁, L₂, L₃, such that |L_i∩L_j|≤k−2(1≤i<j≤3), then G contains a K_k+2 as a minor. Note that a graph G is t-apex if G−X is planar for some subset X⊆V(G) of order at most t.This theorem generalizes some earlier results by Robertson, Seymour and Thomas [N. Robertson, P.D. Seymour, R. Thomas, Hadwiger conjecture for K₆-free graphs, Combinatorica 13 (1993) 279-361.], Kawarabayashi and Toft [K. Kawarabayashi, B. Toft, Any 7-chromatic graph has K₇ or K_4,4 as a minor, Combinatorica 25 (2005) 327-353] and Kawarabayashi, Luo, Niu and Zhang [K. Kawarabayashi, R. Luo, J. Niu, C.-Q. Zhang, On structure of k-connected graphs without K_k-minor, Europ. J. Combinatorics 26 (2005) 293-308].     

Triangulated Spheres and Colored Cliques

Abstract. In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S ^d-1 can be extended to a triangulation of the ball B ^d , by adding one point at a time, having degree at most 2d to its predecessors.     

Hiding Cliques for Cryptographic Security

We demonstrate how a well studied combinatorial optimizationproblem may be used as a new cryptographic primitive. The problemin question is that of finding a "large" clique in a randomgraph. While the largest clique in a random graph with nvertices and edge probability p is very likely tobe of size about , it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size with significantprobability for any constant > 0. We presenta very simple method of exploiting this conjecture by hidinglarge cliques in random graphs. In particular, we show that ifthe conjecture is true, then when a large clique—of size,say, is randomlyinserted (hidden) in a random graph, finding a clique ofsize remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to . Our result suggests several cryptographicapplications, such as a simple one-way function.     

Vertex Covers by Edge Disjoint Cliques

Dedicated to the memory of Paul Erdős Let H be a simple graph having no isolated vertices. An (H,k)-vertex-cover of a simple graph G = (V,E) is a collection of subgraphs of G satisfying 1.  , for all i = 1, ..., r, 2.  , 3.  , for all , and 4.  each is in at most k of the . We consider the existence of such vertex covers when H is a complete graph, , in the context of extremal and random graphs. Received October 31, 1999 RID="*" ID="*" Supported in part by NSF grant DMS-9627408. RID="†" ID="†" Supported in part by NSF grant CCR-9530974. RID="‡" ID="‡" Supported in part by OTKA Grants T 030059 and T 29074, FKFP 0607/1999 and by the Bolyai Foundation. RID="§" ID="§" Supported in part by NSF grant DMS-9970622.     

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

Automorphisms and Distinguishing Numbers of Geometric Cliques

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second, if the boundary of the convex hull contains at least four vertices, then it is invariant under every geometric automorphism. We use these results, and the theory of determining sets, to prove that every geometric n-clique in which n≥7 and the boundary of the convex hull contains at least four vertices is 2-distinguishable.     

Large Scale Oscillations in the Continuous Casting Process

Long-term oscillations of the flow in a continuous casting tundish are investigated numerically and experimentally. The numerical model is based on the Reynolds averaged Navier-Stokes equations (URANS). A comparison between numerical results and experimental data from a water model is presented. Frequencies due to long-term oscillations are resolved in the simulation. These frequencies are in satisfactory agreement with the experimental results of LDA measurements. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finding All Maximal Cliques in Dynamic Graphs

Clustering applications dealing with perception based or biased data lead to models with non-disjunct clusters. There, objects to be clustered are allowed to belong to several clusters at the same time which results in a fuzzy clustering. It can be shown that this is equivalent to searching all maximal cliques in dynamic graphs like G _t = (V,E _t), where E _{t – 1} E _t, t = 1,...,T; E ₀ = . In this article algorithms are provided to track all maximal cliques in a fully dynamic graph.     

Maximum Cliques of Hypergraphs and Polynomial Optimization

A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Buló and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local (global) minimizers of a homogeneous polynomial function of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Turán densities of complete non-uniform hypergraphs.