Complete Bipartite Subgraphs Research Articles

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.

Extremal K(s,t)-free bipartite graphs

Abstract We approach a well-known topic in extremal graph theory, the so-called Zarankiewicz Problem [K. Zarankiewicz, Problem P 101, Colloq. Math. 2 (1951), 301], which consists in: (i) calculating the maximum number of edges z ( m , n ; s , t ) that a bipartite graph G with partite classes of cardinalities m and n can have such that G is free of a complete bipartite subgraph K ( s , t ) with s vertices in the m class and t vertices in the n class; (ii) describing all the corresponding extremal bipartite graphs having that maximum number of edges. In this paper, the exact value of z ( m , n ; s , t ) is calculated and the corresponding family Z ( m , n ; s , t ) of extremal graphs is characterized when the parameters satisfy certain relationships.

Fractional Biclique Covers and Partitions of Graphs

A biclique is a complete bipartite subgraph of a graph. This paper investigates the fractional biclique cover number, $bc^*(G)$, and the fractional biclique partition number, $bp^*(G)$, of a graph $G$. It is observed that $bc^*(G)$ and $bp^*(G)$ provide lower bounds on the biclique cover and partition numbers respectively, and conditions for equality are given. It is also shown that $bc^*(G)$ is a better lower bound on the Boolean rank of a binary matrix than the maximum number of isolated ones of the matrix. In addition, it is noted that $bc^*(G) \leq bp^*(G) \leq \beta^*(G)$, the fractional vertex cover number. Finally, the application of $bc^*(G)$ and $bp^*(G)$ to two different weak products is discussed.

Chromatic characterization of biclique covers

A biclique B of a simple graph G is the edge-set of a complete bipartite subgraph of G . A biclique cover of G is a collection of bicliques covering the edge-set of G. Given a graph G, we will study the following problem: find the minimum number of bicliques which cover the edge-set of G. This problem will be called the minimum biclique cover problem (MBC). First, we will define the families of independent and dependent sets of the edge-set E ( G ) of G: F ⊆ E ( G ) will be called independent if there exists a biclique B ⊆ E ( G ) such that F ⊆ B , and will be called dependent otherwise. From our study of minimal dependent sets we will derive a 0–1 linear programming formulation of the following problem: find the maximum weighted biclique in a graph. This formulation may have an exponential number of constraints with respect to the number of nodes of G but we will prove that the continuous relaxation of this integer program can be solved in polynomial time. Finally we will also study continuous relaxation methods for the problem (MBC). This research was motivated by an open problem of Fishburn and Hammer.

On Sampling Colorings of Bipartite Graphs

We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.

Bisplit graphs

An undirected graph G = ( V , E ) is a bisplit graph if its vertex set can be partitioned into a stable set and a complete bipartite graph. We provide an O ( | V | | E | ) time recognition algorithm for these graphs and characterize them in terms of forbidden induced subgraphs. We also discuss the problem of recognizing whether G has a stable set S such that the connected components of G [ V ⧹ S ] are more than one complete bipartite subgraph.

Confluent Layered Drawings

We combine the idea of confluent drawings with Sugiyama-style drawings in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed-style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge-coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.

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.

Biclique comparability digraphs of bipartite graphs and minimum ranks of partial matrices

Every bipartite graph has a biclique comparability digraph whose vertices are the inclusion-maximal complete bipartite subgraphs of the bipartite graph and whose arcs correspond to inclusions of the relevant color classes. I characterize those digraphs that correspond to bipartite graphs and, in particular, those that correspond to chordal bipartite graphs. This is motivated by work on finding the minimum rank of completions of partially specified matrices. In particular, Woerdeman (Integral Equations Operator Theory 10 (1987) 859) proved a formula for minimum rank in special cases that can be naturally reformulated in terms of the biclique comparability digraphs of the bipartite graphs that have the partial matrices as incidence matrices. Cohen et al. (Oper. Theory Adv. Appl. 40 (1989) 165) conjecture that this formula actually gives the minimum rank if and only if the corresponding bipartite graph is chordal bipartite.

Decompositions of complete multigraphs derived from Hadamard matrices

Let b( μK v ) be the minimum number of complete bipartite subgraphs needed to partition the edge set of μK v , the complete multigraph with μ edges between each pair of vertices. Previous work by Gregory and Vander Meulen determined that for μ odd with v⩽2 μ, and subject to the existence of certain Hadamard and conference matrices, then b( μK v ) is one of two numbers. By considering forbidden submatrices of a vertex–biclique incidence matrix, we determine conditions for when the lower of these numbers is not attained, and describe constructions that show the lower bound can be attained in the remaining cases. Assuming the standard necessary conditions for the existence of Hadamard and conference matrices are sufficient, this completes the determination of b( μK v ) for all μ and v such that v⩽2 μ.

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.

Bicliques and Eigenvalues

A biclique in a graph Γ is a complete bipartite subgraph of Γ. We give bounds for the maximum number of edges in a biclique in terms of the eigenvalues of matrix representations of Γ. These bounds show a strong similarity with Lovász's bound ϑ(Γ) for the Shannon capacity of Γ. Motivated by this similarity we investigate bicliques and the bounds in certain product graphs.

Contributions to the problem of Zarankiewicz

Let G 2 ( n ) denote a bipartite graph with n vertices in each color class, and let z ( n , t ) be the bipartite Turán number, representing the maximum possible number of edges in G 2 ( n ) if it is not to contain a copy of the complete bipartite subgraph K ( t , t ). It is then clear that ζ ( n , t )= n 2 − z ( n , t ) denotes the smallest possible number of zeros in an n × n zero–one matrix that does not contain a t × t submatrix consisting of all ones. We investigate the behaviour of z ( n , t ) when n goes to infinity (and perhaps t →∞ as well). The case t= o (n 1/3 );t⪢ log n has been considered by Godbole et al. [Electron. J. Combin. 4 (1997) 14pp], and we focus here on the overlapping case 2⩽ t ⪡ n 1/5 . Fill an n × n matrix randomly with z ones and n 2 − z = ζ zeros. Then, we prove that if t →∞, the asymptotic probability that there are no t × t submatrices with all ones is one or zero, according as z=n 2−2/t t e 2/t {1+ o (1)} or z=n 2−2/t t e 2/t {1+ o ∗ (1)}, where the o(1) and o ∗ (1) functions may be taken to be ( log t−a n )/t 2 and ( log t+a n )/t 2 respectively, for any a n that tends to infinity. If t is finite, a less sharp result is shown to be valid. The proof employs the Janson exponential inequalities. Finally, the Stein–Chen method is employed to derive a Poisson approximation for the probability distribution of the number of copies of K ( t , t ) in the random bipartite graph; an extreme value limit for the probability that there are no t × t submatrices with all ones, when the value of z is at the threshold level, is derived as a corollary.

A tree whose complement is not eigensharp

The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) is at least the maximum of the number of positive and negative eigenvalues of the adjacency matrix A of G; that is b(G)⩾ max{n +(A),n −(A)} . When equality is attained G is said to be eigensharp. Using known necessary conditions for equality, it is shown that there is a tree on 14 vertices whose complement is not eigensharp. It is also shown that G is the eigensharp when G is the complement of a forest where each component is a path.

Bicliques in Graphs I: Bounds on Their Number

are inclusion-maximal induced complete bipartite subgraphs in graphs. Upper bounds on the number of bicliques in bipartite graphs and general graphs are given. Then those classes of graphs where the number of bicliques is polynomial in the vertex number are characterized, provided the class is closed under induced subgraphs.

Biclique decompositions and Hermitian rank

The Hermitian rank, h( A), of a Hermitian matrix A is defined and shown to equal max{ n +( A), n −( A)}, the maximum of the numbers of positive and negative eigenvalues of A. Properties of Hermitian rank are developed and used to obtain results on the minimum number, b( G), of complete bipartite subgraphs needed to partition the edge set of a graph G. Witsenhausen's inequality b( G)⩾max{ n +( G), n −( G)} is reproved and conditions necessary for equality to hold are given. The results are then used to estimate b( G) for several classes of graphs. For example, if G is the complement of a path then b(G)=⌊ 2 3 (n−1)⌋ , while if G is the complement of a cycle then b(G)=2⌊ 1 3 (n−1)⌋ or ⌊ 1 3 (2n−1)⌋ .

Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs

A biclique cover (resp. biclique decomposition) of a bipartite graph B is a family of complete bipartite subgraphs of B whose edges cover (resp. partition) the edges of B. The minimum cardinality of a biclique cover (resp. biclique decomposition) is denoted by s-dim( B) (resp. s-part( B)). The decision problems associated with the computation of s-dim and s-part are NP-complete for general bipartite graphs; the decision problem associated to s-dim is NP-complete for bipartite chordal graphs, and polynomial for bipartite distance-hereditary graphs, for bipartite convex graphs and for bipartite C4-free graphs. We show here that for bipartite domino-free graphs (a strict generalization of bipartite distance-hereditary graphs and bipartite C4-free graphs), s-dim and s-part are equal and can be computed in O(n × m) time. Moreover, we propose a O(n × m) time algorithm to check the domino-free property and to build the Galois lattice of such graphs.

Decompositions of Complete Multigraphs Related to Hadamard Matrices

Let bp(μKv) be the minimum number of complete bipartite subgraphs needed to partition the edge set ofμKv, the complete multigraph withμedges between each pair of itsvvertices. Many papers have examined bp(μKv) forv⩾2μ. For eachμandvwithv⩽2μ, it is shown here that if certain Hadamard and conference matrices exist, then bp(μKv) must be one of two numbers. Also, generalizations to decompositions and covers by completes-partite subgraphs are discussed and connections to designs and codes are presented.

Threshold Functions for the Bipartite Turán Property

Let $G_2(n)$ denote a bipartite graph with $n$ vertices in each color class, and let $z(n,t)$ be the bipartite Turán number, representing the maximum possible number of edges in $G_2(n)$ if it does not contain a copy of the complete bipartite subgraph $K(t,t)$. It is then clear that $\zeta(n,t)=n^2-z(n,t)$ denotes the minimum number of zeros in an $n\times n$ zero-one matrix that does not contain a $t\times t$ submatrix consisting of all ones. We are interested in the behaviour of $z(n,t)$ when both $t$ and $n$ go to infinity. The case $2\le t\ll n^{1/5}$ has been treated elsewhere; here we use a different method to consider the overlapping case $\log n\ll t\ll n^{1/3}$. Fill an $n \times n$ matrix randomly with $z$ ones and $\zeta=n^2-z$ zeros. Then, we prove that the asymptotic probability that there are no $t \times t$ submatrices with all ones is zero or one, according as $z\ge(t/ne)^{2/t}\exp\{a_n/t^2\}$ or $z\le(t/ne)^{2/t}\exp\{(\log t-b_n)/t^2\}$, where $a_n$ tends to infinity at a specified rate, and $b_n\to\infty$ is arbitrary. The proof employs the extended Janson exponential inequalities.

Bipartite dimensions and bipartite degrees of graphs

A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover G's edges. G's bipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d( G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d ⩽ 2, and investigate the forbidden graphs for d ⩽ n that have the fewest vertices. We note that for complete graphs, d( K n ) = [log 2 n], η( K n ) = d( K n ) for n ⩽ 16, and η( K n ) is unbounded. The list of minimal forbidden induced subgraphs for η ⩽ 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices.