Complete Bipartite Subgraphs Research Articles

A critical probability for biclique partition of Gn,p

The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G) with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take G=Gn,p, where p is constant and less than a certain threshold value p0≈0.312. This verifies a conjecture of Chung and Peng for these values of p. We also show that if p0<p<1/2 then bp(Gn,p)=n−(1+Θ(1))α(Gn,p) with high probability.

Perfect state transfer by means of discrete-time quantum walk on the complete bipartite graph

Perfect state transfer has attracted a great deal of attention recently due to its crucial role in quantum communication and scalable quantum computation. In this paper, we propose the perfect state transfer algorithms with a pair of sender-receiver and two pairs of sender-receiver on the complete bipartite graph respectively. The algorithm with a pair of sender-receiver is implemented through discrete-time quantum walk, flexibly setting the coin operators based on the positions of the sender and receiver. The algorithm with two pairs of sender-receiver ensures that the two quantum states are distributed on both sides of the complete bipartite graph during the process, thereby achieving perfect state transfer. In addition, the quantum circuits corresponding to the algorithms are provided. The algorithms can transfer an arbitrary quantum state and can simultaneously transfer two arbitrary quantum states from the senders to the receivers in any case. Moreover, the algorithms are not only applicable to complete bipartite graphs but also to more graph structures with complete bipartite subgraphs, which will provide potential applications for quantum information processing.

Exact values for some unbalanced Zarankiewicz numbers

AbstractFor positive integers , , and , the Zarankiewicz number is defined to be the maximum number of edges in a bipartite graph with parts of sizes and that has no complete bipartite subgraph containing vertices in the part of size and vertices in the part of size . A simple argument shows that, for each , when . Here, for large , we determine the exact value of in almost all of the remaining cases where . We establish a new family of upper bounds on which complement a family already obtained by Roman. We then prove that the floor of the best of these bounds is almost always achieved. We also show that there are cases in which this floor cannot be achieved and others in which determining whether it is achieved is likely a very hard problem. Our results are proved by viewing the problem through the lens of linear hypergraphs and our constructions make use of existing results on edge decompositions of dense graphs.

Wireless random‐access networks with bipartite interference graphs

AbstractWe consider random‐access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph

The Gyárfás-Sumner conjecture says that for every forest H and every integer k, if G is H-free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H, if G is H-free and does not contain Kt,t as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H, the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer d≥2, if G is H-free and does not contain as a subgraph the complete d-partite graph with parts of cardinality t, then its chromatic number is at most polynomial in t.

On some conjectures on biclique graphs

The bicliqueB of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graphKB(G) of a graph G is the graph with the bicliques of G as its vertices where two vertices of KB(G) are adjacent if and only if they have a common vertex in G. In this paper, we disprove a conjecture of Groshaus and Montero concerning Helly property of biclique graphs (Groshaus and Montero, 2021) and two conjectures of Montero concerning vertex removal in biclique graphs (Montero, 2022). In addition, we confirm a conjecture of Montero about the structural characterization of biclique graphs in the same paper.

Finding biclique partitions of co-chordal graphs

The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number of a co-chordal (complementary graph of chordal) graph G=(V,E) is less than the number of maximal cliques (mc) of its complementary graph: A chordal graph Gc=(V,Ec). We first provide a general framework of the “divide and conquer” heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O(|V|(|V|+|Ec|)) is proposed by applying a lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with a size of mc(Gc)−1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc)−2≤bp(G)≤mc(Gc)−1 if G is a split graph.

On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs

Nowadays, graph theory is one of the most exciting fields of mathematics due to the tremendous developments in modern technology, where it is used in many important applications. The orthogonal double cover (ODC) is a branch of graph theory and is considered as a special class of graph decomposition. In this paper, we decompose the complete bipartite graphs Kx,x by caterpillar graphs using the method of ODCs. The article also deals with constructing the ODCs of Kx,x by general symmetric starter vectors of caterpillar graphs such as stars–caterpillar, the disjoint copies of cycles–caterpillars, complete bipartite caterpillar graphs, and the disjoint copies of caterpillar paths. We decompose the complete bipartite graph by the complete bipartite subgraphs and by the disjoint copies of complete bipartite subgraphs using general symmetric starter vectors. The advantage of some of these new results is that they enable us to decompose the giant networks into large groups of small networks with the comprehensive coverage of all parts of the giant network by using the disjoint copies of symmetric starter subgraphs. The use case of applying the described theory for various applications is considered.

Biclique coloring game (Brief Announcement)

A biclique q-coloring is an assignment of q-colors to the vertices of a graph G, so that no biclique (maximal set of vertices that induces a complete bipartite subgraph of G with at least one edge) is monochromatic. Inspired by the coloring game, we introduce the biclique q-coloring game played on a graph G defined as follows. Two players, Alice and Bob, alternately color the vertices of a graph G using q colors. Alice's goal is to color the vertices of G so that no biclique is monochromatic, and Bob tries to prevent this. Both players play optimally and respect the following rule: if a biclique is fully colored, then there exist at least two vertices in the biclique with different colors. In this paper, we prove that the biclique q-coloring game is PSPACE-complete and study the game in powers of paths Pkn.

Vertex removal in biclique graphs

A biclique is a maximal induced complete bipartite subgraph. The biclique graph of a graph H , denoted by K B ( H ) , is the intersection graph of the family of all bicliques of H . In this work we address the following question: Given a biclique graph G = K B ( H ) , is it possible to remove a vertex q of G , such that G − { q } is a biclique graph? And if possible, can we obtain a graph H ′ such that G − { q } = K B ( H ′ ) ? We show that the general question has a “no” for answer. However, we prove that if G has a vertex q such that d ( q ) = 2 , then G − { q } is a biclique graph and we show how to obtain H ′ .

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.

Well-partitioned chordal graphs

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

A stability theorem for maximal C2k+1 ${C}_{2k+1}$‐free graphs

AbstractFor any positive integer , we show that every maximal ‐free graph with at least edges contains an induced complete bipartite subgraph on vertices. We also show that this is the best possible.

Synthesis of combinational circuits by means of bi-decomposition of Boolean functions

O b j e c t i v e s . The problem of synthesis of combinational circuits in the basis of two-input gates is considered. Those gates are AND, OR, NAND and NOR. The objective of the paper is to investigate the possibilities of application of bi-decomposition of Boolean functions to the synthesis of combinational circuits.M e t h o d s . The method for bi-decomposition is reduced to the search in a graph for a weighted two-block cover with complete bipartite subgraphs (bi-cliques).R e s u l t s . The initial Boolean function is given as two ternary matrices, one of which represents the domain of Boolean space where the function has the value 1, and the other is the domain of Boolean space where the function has the value 0. The orthogonality graph of rows of ternary matrices representing the given function is considered. The method for two-bi-clique covering the orthogonality graph of rows of ternary matrices is described. Every bi-clique in the obtained cover is assigned in a certain way with а set of variables that are the arguments of the function. This set is the weight of the bi-clique. Each of those bi-cliques defines a Boolean function whose arguments are the variables assigned to it. The functions obtained in such a way constitute the required decomposition.Co n c l u s i o n . The process of synthesis of a combinational circuit consists of a successive application of bi-decomposition to obtained functions. The suggested method allows obtaining the circuits with short delay.

Энергосберегающее кодирование частичных состояний параллельного автомата

The problem of a low-power assignment of the partial states of a parallel automaton is considered. A method to solve that problem is suggested that provides minimizing the number of memory elements in the implementing circuit of the automaton and minimization of their switching activity. The problem is reduced to finding a minimal weighted cover of a graph with its complete bipartite sub-graphs (bi-cliques).

Regarding Two Conjectures on Clique and Biclique Partitions

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

On the edge‐biclique graph and the iterated edge‐biclique operator

AbstractA biclique of a graph is a maximal induced complete bipartite subgraph of . The edge‐biclique graph of , , is the edge‐intersection graph of the bicliques of . A graph diverges (resp. converges or is periodic) under an operator whenever (resp. for some or for some and ). The iterated edge‐biclique graph of , , is the graph obtained by applying the edge‐biclique operator successive times to . In this article, we first study the connectivity relation between and . Next, we study the iterated edge‐biclique operator . In particular, we give sufficient conditions for a graph to be convergent or divergent under the operator , we characterize the behavior of burgeon graphs and we propose some general conjectures on the subject.

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.

The biclique partitioning polytope

Given a bipartite graph G=(V1,V2,E), a biclique of G is a complete bipartite subgraph of G, and a biclique partitioning of G is a set A⊆E such that the bipartite graph G′=(V1,V2,A) is a vertex-disjoint union of bicliques. The biclique partitioning problem (BPP) consists of, given a complete bipartite graph G=(V1,V2,E) with edge weights we∈R for all e∈E (thus, negative weights are allowed), finding a biclique partitioning A⊆E of minimum weight. The bicluster editing problem (BEP) is a variant of the BPP and consists of editing a minimum number of edges of an input bipartite graph G in order to transform its edge set into a biclique partitioning. Editing an edge consists of either adding it to the graph or deleting it from the graph. In addition to the BPP and the BEP, other problems in the literature aim at finding biclique partitionings, and this motivates the study of the biclique partitioning polytope Pnm of the complete bipartite graph Knm (i.e., Pnm is the convex hull of the incidence vectors of the biclique partitionings of Knm). In this paper we develop such a polyhedral study and show that ladder, bellows, and grid inequalities induce facets of Pnm. Our computational results show that these inequalities are very effective in solving the BEP. In particular, they are able to improve the value of the relaxed solution by up to 20%.

Recognizing Generating Subgraphs Revisited

A graph [Formula: see text] is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function [Formula: see text] is defined on its vertices. Then [Formula: see text] is [Formula: see text]well-covered if all maximal independent sets are of the same weight. For every graph [Formula: see text], the set of weight functions [Formula: see text] such that [Formula: see text] is [Formula: see text]-well-covered is a vector space, denoted as WCW(G). Deciding whether an input graph [Formula: see text] is well-covered is co-NP-complete. Therefore, finding WCW(G) is co-NP-hard. A generating subgraph of a graph [Formula: see text] is an induced complete bipartite subgraph [Formula: see text] of [Formula: see text] on vertex sets of bipartition [Formula: see text] and [Formula: see text], such that each of [Formula: see text] and [Formula: see text] is a maximal independent set of [Formula: see text], for some independent set [Formula: see text]. If [Formula: see text] is generating, then [Formula: see text] for every weight function [Formula: see text]. Therefore, generating subgraphs play an important role in finding WCW(G). The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article we prove NP- completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other hand, we supply polynomial algorithms for recognizing generating subgraphs and finding WCW(G), when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds WCW(G) when [Formula: see text] does not contain cycles of lengths 3, 4, 5, and 7.