Largest Clique Research Articles

Fractional clique decompositions of dense graphs and hypergraphs

Our main result is that every graph G on n≥104r3 vertices with minimum degree δ(G)≥(1−1/104r3/2)n has a fractional Kr-decomposition. Combining this result with recent work of Barber, Kühn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F-decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant ck>0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least (1−ck/r2k−1)n has a fractional Kr(k)-decomposition, where Kr(k) is the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large F-divisible complete graph has an F-decomposition.

Dense 3-uniform hypergraphs containing a large clique

An r-uniform graph G is dense if and only if every proper subgraph G′ of G satisfies λ(G′) < λ(G), where λ(G) is the Lagrangian of a hypergraph G. In 1980’s, Sidorenko showed that π(F), the Turan density of an r-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all dense F-hom-free r-uniform hypergraphs. This connection has been applied in the estimating Turan density of hypergraphs. When r = 2, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when r ≥ 3, it becomes much harder to estimate the Lagrangians of r-uniform hypergraphs and to characterize the structure of all dense r-uniform graphs. The main goal of this note is to give some sufficient conditions for 3-uniform graphs with given substructures to be dense. For example, if G is a 3-graph with vertex set [t]and m edges containing [t − 1](3), then G is dense if and only if $$m \geqslant \left( {\frac{{t - 1}}{3}} \right) + \left( {\frac{{t - 2}}{2}} \right) + 1$$ . We also give a sufficient condition on the number of edges for a 3-uniform hypergraph containing a large clique minus 1 or 2 edges to be dense.

Large Cliques in Sparse Random Intersection Graphs

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish the asymptotic order of the clique number $\omega(G)$ of a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha&gt;2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.

Large Subgraphs in Rainbow‐Triangle Free Colorings

AbstractFox–Grinshpun–Pach showed that every 3‐coloring of the complete graph on n vertices without a rainbow triangle contains a clique of size that uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such coloring contains a 2‐colored subgraph with chromatic number at least , and this is best possible. We further show that for fixed positive integers with , every r‐coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a subgraph that uses at most s colors and has chromatic number at least , and this is best possible. Fox–Grinshpun–Pach previously showed a clique version of this result. As a direct corollary of our result we obtain a generalization of the celebrated theorem of Erdős‐Szekeres, which states that any sequence of n numbers contains a monotone subsequence of length at least . We prove that if an r‐coloring of the edges of an n‐vertex tournament does not contain a rainbow triangle then there is an s‐colored directed path on vertices, which is best possible. This gives a partial answer to a question of Loh.

A proof of Mader's conjecture on large clique subdivisions in C4-free graphs

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$.

Colouring perfect graphs with bounded clique number

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver [9] from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a “combinatorial” polynomial-time algorithm that yields an optimal colouring of a perfect graph.A skew partition in G is a partition (A,B) of V(G) such that G[A] is not connected and G‾[B] is not connected, where G‾ denotes the complement graph; and it is balanced if an additional parity condition on certain paths in G and G‾ is satisfied.In this paper we first give a polynomial-time algorithm that, with input a perfect graph, outputs a balanced skew partition if there is one. Then we use this to obtain a combinatorial algorithm that finds an optimal colouring of a perfect graph with clique number k, in time that is polynomial for fixed k.

A fast space-saving algorithm for maximal co-location pattern mining

Real space teems with potential feature patterns with instances that frequently appear in the same locations. As a member of the data-mining family, co-location can effectively find such feature patterns in space. However, given the constant expansion of data, efficiency and storage problems become difficult issues to address. Here, we propose a maximal-framework algorithm based on two improved strategies. First, we adopt a degeneracy-based maximal clique mining method to yield candidate maximal co-locations to achieve high-speed performance. Motivated by graph theory with parameterized complexity, we regard the prevalent size-2 co-locations as a sparse undirected graph and subsequently find all maximal cliques in this graph. Second, we introduce a hierarchical verification approach to construct a condensed instance tree for storing large instance cliques. This strategy further reduces computing and storage complexities. We use both synthetic and real facility data to compare the computational time and storage requirements of our algorithm with those of two other competitive maximal algorithms: “order-clique-based” and “MAXColoc”. The results show that our algorithm is both more efficient and requires less storage space than the other two algorithms.

The typical structure of graphs with no large cliques

In 1987, Kolaitis, Promel and Rothschild proved that, for every fixed r∈N, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r ≤ (logn)1/4. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobas and Simonovits.

Approximation of the Quadratic Knapsack Problem

We study the approximability of the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are not given for each pair of knapsack items. Instead, an edge weighted graph, whose vertices represent the knapsack items, induces a quadratic profit for every pair of items, which is adjacent in the graph. We show that the problem permits an FPTAS on graphs of bounded treewidth and a PTAS on planar graphs and more generally on H-minor free graphs. We also show strong 𝒩𝒫-hardness of QKP on graphs that are 3-book embeddable, a natural graph class that is related to planar graphs. In addition, we will argue that the problem is likely to have bad approximability behaviour on all graph classes that include the complete graph or contain large cliques. These hardness of approximation results under certain complexity assumptions carry over from the densest k-subgraph problem.

Edge-decompositions of graphs with high minimum degree

A fundamental theorem of Wilson states that, for every graph F, every sufficiently large F-divisible clique has an F-decomposition. Here a graph G is F-divisible if e(F) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F-decomposition if the edges of G can be covered by edge-disjoint copies of F. We extend this result to graphs G which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3-divisible graph of minimum degree at least 9n/10+o(n) has a K3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3-divisible graph with minimum degree at least 3n/4 has a K3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3+o(n) for the existence of a C4-decomposition, and of n/2+o(n) for the existence of a C2ℓ-decomposition, where ℓ≥3. Our main contribution is a general ‘iterative absorption’ method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3-divisible graph with minimum degree at least 3n/4+o(n) has an approximate K3-decomposition.

Communication cliques in mobile phone calling networks

.People in modern societies form different social networks through numerous means of communication. These communication networks reflect different aspects of human’s societal structure. The billing records of calls among mobile phone users enable us to construct a directed calling network (DCN) and its Bonferroni network (SVDCN) in which the preferential communications are statistically validated. Here we perform a comparative investigation of the cliques of the original DCN and its SVDCN constructed from the calling records of more than nine million individuals in Shanghai over a period of 110 d. We find that the statistical properties of the cliques of the two calling networks are qualitatively similar and the clique members in the DCN and the SVDCN exhibit idiosyncratic behaviors quantitatively. Members in large cliques are found to be spatially close to each other. Based on the clique degree profile of each mobile phone user, the most active users in the two calling networks can be classified into several groups. The users in different groups are found to have different calling behaviors. Our study unveils interesting communication behaviors among mobile phone users that are densely connected to each other.

Edge-decompositions of graphs with high minimum degree

A fundamental theorem of Wilson states that, for every graph F, every sufficiently large F-divisible clique has an F-decomposition. Here a graph G is F-divisible if e(F) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F-decomposition if the edges of G can be covered by edge-disjoint copies of F. We extend this result to graphs which are allowed to be far from complete: our results imply that every sufficiently large F-divisible graph G on n vertices with minimum degree at least (1−1/(16|F|4)+ε)n has an F-decomposition. Moreover, every sufficiently large K3-divisible graph of minimum degree at least 0.956n has a K3-decomposition. Our result significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3-divisible graph with minimum degree 3n/4 has a K3-decomposition. For certain graphs, we can strengthen the general bound above. In particular, we obtain the asymptotically correct thresholds of 2n/3+o(n) for C4 and n/2+o(n) for even cycles of length at least 6. Our main contribution is a general method which turns an approximate decomposition into an exact one.

Subdivisions of a large clique in [formula omitted]-free graphs

Mader conjectured that every C4-free graph has a subdivision of a clique of order linear in its average degree. We show that every C6-free graph has such a subdivision of a large clique.We also prove the dense case of Mader's conjecture in a stronger sense, i.e., for every c, there is a c′ such that every C4-free graph with average degree cn1/2 has a subdivision of a clique Kℓ with ℓ=⌊c′n1/2⌋ where every edge is subdivided exactly 3 times.

Mid size cliques are more common in real world networks than triangles

AbstractReal world networks typically have large clustering coefficients. The clustering coefficient can be interpreted to be the result of a triangle closing mechanism. We have here enumerated cliques and maximal cliques in multiple networks to show that real world networks have a high number of large cliques. While triangles are more frequent than expected, large cliques are much more over-expressed, and the largest difference between real world networks and their random counterpart occurs in many networks at clique sizes of 5–7, and not at a size of 3. This does not result from the existence of few very large cliques, since a similar feature is observed when studying only maximal cliques (cliques that are not contained in other larger cliques). Moreover, when the large cliques are removed, triangles are often under-expressed.In all networks studied but one, all node members of large cliques produce a single connected component, which represent the central “core” of the network. The observed clique distribution can be explained by multiple models, mainly hidden variables model, such as the gravitation model, or the collapse of bipartite networks. These models can explain other properties of these networks, including the sub-graph distribution and the distance distribution of the networks. This suggests that node connectivity in real world networks may be determined by the similarity between the contents of the networks' nodes. This is in contrast with models of network formation that incorporate only the properties of the network, and not the internal properties of the nodes.

The Erdős–Hajnal conjecture for rainbow triangles

We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Ω(n1/3log2⁡n) which uses at most two colors, and this bound is tight up to a constant factor. This verifies a conjecture of Hajnal which is a case of the multicolor generalization of the well-known Erdős–Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s≤r, we determine a constant cr,s such that the following holds. Every r-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Ω(ns(s−1)/r(r−1)(log⁡n)cr,s) which uses at most s colors, and this bound is tight apart from the implied constant factor. The proof of the lower bound utilizes Gallai's classification of rainbow-triangle free edge-colorings of the complete graph, a new weighted extension of Ramsey's theorem, and a discrepancy inequality in edge-weighted graphs. The proof of the upper bound uses Erdős' lower bound on Ramsey numbers by considering lexicographic products of 2-edge-colorings of complete graphs without large monochromatic cliques.

Statistical Inverse Ray Tracing for Image-Based 3D Modeling.

This paper proposes a new formulation and solution to image-based 3D modeling (aka "multi-view stereo") based on generative statistical modeling and inference. The proposed new approach, named statistical inverse ray tracing, models and estimates the occlusion relationship accurately through optimizing a physically sound image generation model based on volumetric ray tracing. Together with geometric priors, they are put together into a Bayesian formulation known as Markov random field (MRF) model. This MRF model is different from typical MRFs used in image analysis in the sense that the ray clique, which models the ray-tracing process, consists of thousands of random variables instead of two to dozens. To handle the computational challenges associated with large clique size, an algorithm with linear computational complexity is developed by exploiting, using dynamic programming, the recursive chain structure of the ray clique. We further demonstrate the benefit of exact modeling and accurate estimation of the occlusion relationship by evaluating the proposed algorithm on several challenging data sets.

An integer programming approach for finding the most and the least central cliques

We consider the problem of finding the most and the least “influential” or “influenceable” cliques in graphs based on three classical centrality measures: degree, closeness and betweenness. In addition to standard clique betweenness, which is defined as the proportion of shortest paths between any two graph nodes that pass through the clique, we also consider its optimistic and pessimistic versions, where outside nodes may favor or try to avoid shortest paths passing through the clique, respectively. We discuss the computational complexity issues for these problems and develop linear 0–1 programming formulations for each centrality metric. Finally, we demonstrate the performance of the developed formulations on real-life and synthetic networks, and provide some interesting insights based on the obtained results. In particular, our findings indicate that there are considerable variations between the centrality values of large cliques within the same networks. Moreover, the most central cliques in graphs are not necessarily the largest ones.

The edge-disjoint paths problem in Eulerian graphs and 4-edge-connected graphs

In the edge-disjoint paths problem, we are given a graph and a set of k pairs of vertices, and we have to decide whether or not the graph has k edge-disjoint paths connecting given pairs of terminals. Robertson and Seymour's graph minor project gives rise to a polynomial time algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project. We give a faster algorithm and a much simpler proof of the correctness for the edge-disjoint paths problem. Our results can be summarized as follows: 1.If an input graph is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤3-edge-cuts. (iii) Excluding large clique minors.2.When an input graph is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be a non-trivially super constant number, up to k=O((log log logn)½??) for any ? > 0. In addition, if an input graph is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((logn½??) for any ? > 0.3.We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow the Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, our algorithm is faster than Robertson and Seymour's.

The Topology of Competitively Constructed Graphs

We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.

Coloring the edges of a directed graph

Finding large cliques in a graph is an important problem in applied discrete mathematics. In directed graph a possible corresponding problem is finding large transitive subtournaments. It is well-known that coloring the graph speeds up the clique search in the undirected case. In this paper we propose coloring schemes to speed up the tournament search in the directed case. We prove two complexity results about the proposed colorings. A consequence of these results is that in practical computations we have to be content with approximate greedy coloring algorithms.