Largest Complete Subgraph Research Articles

Properly colored cycles in edge-colored 2-colored-triangle-free complete graphs

An edge-colored graph Gc is called properly colored if any two adjacent edges receive distinct colors. An edge-colored graph Gc is 2-colored-triangle-free if Gc contains no 2-colored-triangle, where a 2-colored-triangle is an edge-colored triangle with exactly two colors. Let dc(v) be the number of colors on the edges incident to v in Gc and let δc(Gc) be the minimum dc(v) for all v∈V(Gc). In this paper we extend the definitions of proper vertex-pancyclic and proper edge-pancyclic to proper k-path-pancyclic, defined as follows: An edge-colored graph Gc is said to be proper k-path-pancyclic if each properly colored path of length k is contained in a properly colored cycle of length l for every l with max{3,k+2}≤l≤|V(Gc)|. We prove that an edge-colored 2-colored-triangle-free complete graph Gc with δc(Gc)≥k+3 is either (almost) proper k-path-pancyclic or contains a large monochromatic complete subgraph.

Point-hyperplane Incidence Geometry and the Log-rank Conjecture

We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in ℝ d that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., 2 –polylog( d ) ) fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. In other words, the point-hyperplane incidence graph for such configurations has a large complete bipartite subgraph. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank- d matrix containing at most O (1) distinct entries in each column contains a submatrix of fractional size 2 –polylog( d ) , in which each column is constant. We prove that our conjecture is equivalent to the log-rank conjecture; the crucial ingredient of this proof is a reduction from bounds for parallel k -partitions to bounds for parallel ( k -1)-partitions. We also introduce an (apparent) strengthening of the conjecture, which relaxes the requirements that the sets of hyperplanes be parallel. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density Ω (ε 2 d / d ) in any d -dimensional configuration with incidence density ε, qualitatively matching previous results proved using sophisticated geometric techniques. We also improve an upper-bound construction of Apfelbaum and Sharir [ 2 ], yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is Ω (1/√ d ). Finally, we discuss various constructions (due to others) of products of Boolean matrices which yield configurations with incidence density Ω (1) and complete bipartite subgraph density 2 -Ω (√ d ) , and pose several questions for this special case in the alternative language of extremal set combinatorics. Our framework and results may help shed light on the difficulty of improving Lovett’s Õ(√ rank( f )) bound [ 20 ] for the log-rank conjecture. In particular, any improvement on this bound would imply the first complete bipartite subgraph size bounds for parallel 3-partitioned configurations which beat our generic bounds for unstructured configurations.

On-line interval graphs coloring — Modification of the First-Fit algorithm and its performance ratio

We propose a modification of the First-Fit algorithm in which we forbid the use of the most frequently used color. We provide some lower bound of the performance ratio [Formula: see text] on the family of interval graphs [Formula: see text], where [Formula: see text] denotes the number of used colors by the modified version of the First-Fit algorithm and [Formula: see text] is the number of vertices in the largest complete subgraph in [Formula: see text]. We also compare the modification with the usual First-Fit algorithm on some experimental data.

Degeneracy of Pt-free and C⩾t-free graphs with no large complete bipartite subgraphs

A hereditary class of graphs G is χ-bounded if there exists a function f such that every graph G∈G satisfies χ(G)⩽f(ω(G)), where χ(G) and ω(G) are the chromatic number and the clique number of G, respectively. As one of the first results about χ-bounded classes, Gyárfás proved in 1985 that if G is Pt-free, i.e., does not contain a t-vertex path as an induced subgraph, then χ(G)⩽(t−1)ω(G)−1. In 2017, Chudnovsky, Scott, and Seymour proved that C⩾t-free graphs, i.e., graphs that exclude induced cycles with at least t vertices, are χ-bounded as well, and the obtained bound is again superpolynomial in the clique number. Note that Pt−1-free graphs are in particular C⩾t-free. It remains a major open problem in the area whether for C⩾t-free, or at least Pt-free graphs G, the value of χ(G) can be bounded from above by a polynomial function of ω(G). We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every t there exists a constant c such that for every ℓ and every C⩾t-free graph which does not contain Kℓ,ℓ as a subgraph, it holds that χ(G)⩽ℓc.

Discovering the maximum k-clique on social networks using bat optimization algorithm

The k-clique problem is identifying the largest complete subgraph of size k on a network, and it has many applications in Social Network Analysis (SNA), coding theory, geometry, etc. Due to the NP-Complete nature of the problem, the meta-heuristic approaches have raised the interest of the researchers and some algorithms are developed. In this paper, a new algorithm based on the Bat optimization approach is developed for finding the maximum k-clique on a social network to increase the convergence speed and evaluation criteria such as Precision, Recall, and F1-score. The proposed algorithm is simulated in Matlab® software over Dolphin social network and DIMACS dataset for k = 3, 4, 5. The computational results show that the convergence speed on the former dataset is increased in comparison with the Genetic Algorithm (GA) and Ant Colony Optimization (ACO) approaches. Besides, the evaluation criteria are also modified on the latter dataset and the F1-score is obtained as 100% for k = 5.

Improved bounds for the Erdős-Rogers function

[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{\alpha_s}$ vertices with $\alpha_s<1/2$ for every $s\ge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2\le t\le 2s-1$ and $s\ge 3$.

A Ramsey Theorem for Biased Graphs

A biased graph is a pair $(G,\mathcal{B})$, where $G$ is a graph and $\mathcal{B}$ is a collection of “balanced” cycles of $G$ such that no $\Theta$-subgraph of $G$ contains precisely two balanced cycles. We prove a Ramsey-type theorem, showing that if $(G,\mathcal{B})$ is a biased graph for which $G$ is a very large complete graph, then $G$ contains a large complete subgraph $H$ such that the set of balanced cycles within $H$ has one of three specific, highly symmetric structures, all of which can be described naturally via group-labelings.

The Ramsey number of books

Ramsey's Theorem is among the most well-known results in combinatorics. The theorem states that any two-edge-coloring of a sufficiently large complete graph contains a large monochromatic complete subgraph. Indeed, any two-edge-coloring of a complete graph with N=4t−o(t) vertices contains a monochromatic copy of Kt. On the other hand, a probabilistic argument yields that there exists a two-edge-coloring of a complete graph with N=2t/2+o(t) with no monochromatic copy of Kt. Much attention has been paid to improving these classical bounds but only improvements to lower order terms have been obtained so far. A natural question in this setting is to ask whether every two-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of Kt that can be extended in many ways to a monochromatic copy of Kt+1. Specifically, Erdős, Faudree, Rousseau and Schelp in the 1970's asked whether every two-edge-coloring of KN contains a monochromatic copy of Kt that can be extended in at least (1−ok(1))2−tN ways to a monochromatic copy of Kt+1. A random two-edge-coloring of KN witnesses that this would be best possible. While the intuition coming from random constructions can be misleading, for example, a famous construction by Thomason shows the existence of a two-edge-coloring of a complete graph with fewer monochromatic copies of Kt than a random two-edge-coloring, this paper confirms that the intuition coming from a random construction is correct in this case. In particular, the author answers this question of Erdős et al. in the affirmative. The question can be phrased in the language of Ramsey theory as a problem on determining the Ramsey number of book graphs. A book graph B(k)t is a graph obtained from Kt by adding k new vertices and joining each new vertex to all the vertices of Kt. The main result of the paper asserts that every two-edge-coloring of a complete graph with N=2kt+ok(t) vertices contains a monochromatic copy of B(k)t.

Induced Subgraphs of Graphs With Large Chromatic Number. X. Holes of Specific Residue

A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. As an application, we prove two conjectures of Kalai and Meshulam from the 1990’s connecting the chromatic number of a graph with the homology of its independence complex.

Representation Complexities of SemiAlgebraic Graphs

The representation complexity of a bipartite graph $G=(P,Q)$ is the minimum size $\sum_{i=1}^s (|A_i|+|B_i|)$ over all possible ways to write $G$ as a (not necessarily disjoint) union of complete bipartite subgraphs $G=\cup_{i=1}^s A_i\times B_i$ where $A_i\subset P, B_i\subset Q$ for $i=1,\dots, s$. In this paper we prove that if $G$ is semialgebraic, i.e., when $P$ is a set of $m$ points in $\mathbb{R}^{d_1}$, $Q$ is a set of $n$ points in $\mathbb{R}^{d_2}$, and the edges are defined by some semialgebraic relations, the representation complexity of $G$ is $O( m^{\frac{d_1d_2-d_2}{d_1d_2-1}+\varepsilon} n^{\frac{d_1d_2-d_1}{d_1d_2-1}+\varepsilon}+m^{1+\varepsilon}+n^{1+\varepsilon})$ for arbitrarily small positive $\varepsilon$. This generalizes results by Apfelbaum--Sharir and Solomon--Sharir. As a consequence, when $G$ is $K_{u,u}$-free for some positive integer $u$, its number of edges is $O(u m^{\frac{d_1d_2-d_2}{d_1d_2-1}+\varepsilon} n^{\frac{d_1d_2-d_1}{d_1d_2-1}+\varepsilon}+ u m^{1+\varepsilon}+u n^{1+\varepsilon})$. This bound is stronger than a result of Fox, Pach, Sheffer, Suk, and Zahl when the first term dominates and $u$ grows with $m,n$. Another consequence is that we can find a large complete bipartite subgraph in a semialgebraic graph when the number of edges is large. Similar results hold for semialgebraic hypergraphs.

Large Communities in a Scale-Free Network

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we prove that the random graph $G_{t}$ produced by the so-called GLP model at time $t$ contains a complete subgraph of order $t^\alpha$, where $\alpha = (1-\varepsilon)\frac{1-p}{2-p}$, $\varepsilon$ is any number such that $0<\varepsilon<1$, and $0<p<1$ is a parameter of the model.

Fractional Turan's theorem and bounds for the chromatic number

Turan's theorem gives conditions for finding large complete subgraphs in a graph in terms of the number of edges. In this work we look for families of graphs in which there is a similar theorem, but which will allow us to find larger complete subgraphs. This question is motivated by a geometric result by Katchalski and Liu concerning a fractional Helly theorem.We present results in two directions. On the one hand, we characterize the families of graphs for which there exists a constant β such that a proportion of α edges guarantees the existence of a complete subgraph using a proportion of αβ of the total number of vertices. On the other hand, we give a criterion to guarantee that in a family of graphs any fixed proportion of the edges is sufficient to conclude that the order of the largest complete subgraph goes to infinity as the number of vertices goes to infinity. These results are obtained by giving an upper bound for the chromatic number of a graph in terms of the clique number and an arbitrarily small proportion of the number of vertices.

Dependence Polynomials of some Graph Operations

Suppose G is a simple graph and ck = ck(G) denotes the number of complete subgraphs of size k in G. Then the dependence polynomial of G is defined as fG(x) = 1−c1x+c2x2−c3x3+⋯+(−1)nxn, where n is the size of the largest complete subgraph in G. In this paper, exact formulas for dependence polynomial of some graph operations are presented.

Dynamics of large cooperative pulsed-coupled networks

We study the deterministic dynamics of networks N composed by m non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of m. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" \ p >=1, and that if the cells are mutually structurally identical or similar, then the synchronization is complete (p= 1) . Second, we prove that the amount of information H that N generates or processes equals log p. Therefore, if N completely synchronizes, the information is null. Finally, we prove that N protects the cells from their risk of death.

A continuous characterization of the maximum-edge biclique problem

The problem of finding large complete subgraphs in bipartite graphs (that is, bicliques) is a well-known combinatorial optimization problem referred to as the maximum-edge biclique problem (MBP), and has many applications, e.g., in web community discovery, biological data analysis and text mining. In this paper, we present a new continuous characterization for MBP. Given a bipartite graph $$G$$ , we are able to formulate a continuous optimization problem (namely, an approximate rank-one matrix factorization problem with nonnegativity constraints, R1N for short), and show that there is a one-to-one correspondence between (1) the maximum (i.e., the largest) bicliques of $$G$$ and the global minima of R1N, and (2) the maximal bicliques of $$G$$ (i.e., bicliques not contained in any larger biclique) and the local minima of R1N. We also show that any stationary points of R1N must be close to a biclique of $$G$$ . This allows us to design a new type of biclique finding algorithm based on the application of a block-coordinate descent scheme to R1N. We show that this algorithm, whose algorithmic complexity per iteration is proportional to the number of edges in the graph, is guaranteed to converge to a biclique and that it performs competitively with existing methods on random graphs and text mining datasets. Finally, we show how R1N is closely related to the Motzkin---Strauss formalism for cliques.

Large Complete Bipartite Subgraphs In Incidence Graphs Of Points And Hyperplanes

We show that if the number I of incidences between m points and n planes in $\mathbb{R}^3$ is sufficiently large, then the incidence graph (which connects points to their incident planes) contains a large complete bipartite subgraph involving r points and s planes, so that $rs \ge \frac{I^2}{mn} - a(m+n)$, for some constant $a>0$. This is shown to be almost tight in the worst case because there are examples of arbitrarily large sets of points and planes where the largest complete bipartite incidence subgraph records only $\frac{I^2}{mn}-\frac{m+n}{16}$ incidences. We also take some steps towards generalizing this result to higher dimensions.

The recognizability of sets of graphs is a robust property

Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of context-free sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement ( HR) context-free graph grammars, to Vertex Replacement ( VR) context-free graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifier-free formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HR-recognizability and VR-recognizability coincide. The same combinatorial condition equates HR-context-free and VR-context-free sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures.

On the edge set of graphs of lattice paths

This note explores a new family of graphs defined on the set of paths of the m × n lattice. We let each of the paths of the lattice be represented by a vertex, and connect two vertices by an edge if the corresponding paths share more than k steps, where k is a fixed parameter 0 = k = m + n. Each such graph is denoted by G(m, n, k). Two large complete subgraphs of G(m, n, k) are described for all values of m, n, and k. The size of the edge set is determined for n = 2, and a complicated recursive formula is given for the size of the edge set when k = 1.

The Complexity of the Kn,n-Problem for Node Replacement Graph Languages

The bounded Kn,n-problem is the question whether or not a graph language of a given graph grammar contains arbitrarily large complete bipartite subgraphs Kn,n. In this paper, we investigate the complexity of this problem for all relevant classes of node replacement graph grammers. Our main result states that the bounded Kn,n-problem is NL-complete for reduced nonblocking eNCE graph grammars and for reduced linear NCE graph grammars. As an application, our results settle the complexity of the problems whether or not the graph language of a given confluent, boundary, or linear graph grammar has bounded tree-width and whether or not it is an HR graph language.

On the clique number of a random overlap graph

The behaviour of the size of the largest complete subgraph contained in an overlap graph of a random acyclic digraph is presented.