Maximum Number Of Edges Research Articles

Maximum incomplete recursive circulants in graph embeddings

An incomplete recursive circulant possesses virtually every advantage of a complete recursive circulant, including simple deadlock-free routing, a small diameter, a good support of parallel algorithms, and so on. It is natural to reconfigure a faulty recursive circulant into a maximum incomplete recursive circulant so as to lower potential performance degradation. For [Formula: see text], the maximum incomplete subgraph problem is to identify a subgraph [Formula: see text] of a graph [Formula: see text] on [Formula: see text] vertices having the maximum number of edges among all subgraphs on [Formula: see text] vertices and is NP-complete. In this paper we identify maximum incomplete recursive circulants and use them as a tool to compute the exact wirelength of embedding recursive circulants into special classes of trees, such as [Formula: see text]-rooted complete binary trees, [Formula: see text]-rooted sibling trees, binomial trees, certain caterpillars and path.

On Turán densities of small triple graphs

For a family of k-graphs F, the Turán number T(n,F) is the maximum number of edges in a k-graph of order n that does not contain any member of F. The Turán density t(F)=limn→∞T(n,F)/nk. Let K4 be the tetrahedron that is the complete triple graph of order four. Let the triple graph Fp,q defined on the vertex set P∪Q with |P|=p and |Q|=q consist of those edges which intersect P in either one or three vertices. Let V={1,2,3,4,5} and let F5 be defined on V with E(F5)={123,145,245}. Let F5c denote the complement of F5, and let P5 be the weak pentagon obtained from F5 by adding the edge 134 and let C5 be the pentagon obtained from P5 by adding one more edge 235. We prove that•1/2≤t(F1,4)≤2/3,•23−3≤t(K4,F5c)≤2−2,•1/4≤t(F1,4,P5)≤62−8,•2/9≤t(F1,3,F3,2,C5)≤1/11. The first result relates to a conjecture of Mubayi and Markström–Talbot that t(F1,3)=2/7. The best known bounds are 2/7≤t(F1,3)<0.32908. The second result relates to an old conjecture of Turán that t(K4)=5/9. The best bounds are 5/9≤t(K4)≤(3+17)/12. The last two results relate to a result of Mubayi and Rödl that t(F1,3,C5)≤10/31.

Song Recommendation System Using Maximal b-Matching

The last decade has witnessed a fundamental paradigm shift on how information content is distributed among people. Nowadays, an increasing number of platforms allow everyone to participate both in information production and information consumption. The phenomenon has been coined as democratization of content. However, as the opportunities to find relevant information and relevant audience increases, so does the complexity of a system that would allow suppliers and consumers to meet in the most efficient way. Our motivation is building a “featured item” component for social-media applications. Such a component would provide recommendations to consumers each time they login the system. The existing system follows either collaborative filtering or content based filtering. Collaborative filtering methods are based on collecting and analyzing a large amount of information on user’s behaviours, activities or preferences and predicting what users will like based on their similarity to other users. Content-based filtering methods are based on a description of the item and a profile of the user's preference. Both of these methods require input from the user in the form of ratings or other user's likes. But social content matching takes into account only the user's preferences and also the capacity constraints. For each item 't' and each user 'u', consider constraints on the maximum number of edges that t and u can participate in the matching. These capacity constraints can be estimated by the activity of each user and the relative frequency with which items need to be delivered. Here we introduce the concept called b-matching goal is to find a matching that satisfies all capacity constraints and maximizes the total weight of the edges in the matching. The result of b-matching is the set of songs that are to be recommended to the user based on his likes.

The competition graphs of oriented complete bipartite graphs

In this paper, we study the competition graphs of oriented complete bipartite graphs. We characterize graphs that can be represented as the competition graphs of oriented complete bipartite graphs. We also present the graphs having the maximum number of edges and the graphs having the minimum number of edges among such graphs.

A Density Corrádi–Hajnal Theorem

AbstractWe find, for all sufficiently largenand eachk, the maximum number of edges in ann-vertex graph that does not containk+ 1 vertex-disjoint triangles.This extends a result of Moon [Canad. J.Math. 20 (1968), 96–102], which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corrádi–Hajnal Theorem.

New bounds on the maximum number of edges in k-quasi-planar graphs

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O(n). Fox and Pach showed that every k-quasi-planar graph with n vertices has at most n(log⁡n)O(log⁡k) edges. We improve this upper bound to 2α(n)cnlog⁡n, where α(n) denotes the inverse Ackermann function and c depends only on k, for k-quasi-planar graphs in which any two edges intersect in a bounded number of points. We also show that every k-quasi-planar graph with n vertices in which any two edges have at most one point in common has at most O(nlog⁡n) edges. This improves the previously known upper bound of 2α(n)cnlog⁡n obtained by Fox, Pach, and Suk.

Linear bounds on matrix extremal functions using visibility hypergraphs

The 0–1 matrix A contains a 0–1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeros. If A does not contain M, then A avoids M. Let ex(n,M) be the maximum number of ones in an n×n 0–1 matrix that avoids M, and let exk(m,M) be the maximum number of columns in a 0–1 matrix with m rows that avoids M and has at least k ones in every column. A method for bounding ex(n,M) by using bounds on the maximum number of edges in bar visibility graphs was introduced in Fulek (2009). By using a similar method with bar visibility hypergraphs, we obtain linear bounds on the extremal functions of other forbidden 0–1 matrices.

On the maximal number of edges in a uniform hypergraph with one forbidden intersection

On the maximal number of edges in a uniform hypergraph with one forbidden intersection

An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems

We investigate the computational complexity of the Densestk-Subgraph problem, where the input is an undirected graph G=(V,E) and one wants to find a subgraph on exactly k vertices with the maximum number of edges. We extend previous work on Densestk-Subgraph by studying its parameterized complexity for parameters describing the sparseness of the input graph and for parameters related to the solution size k.On the positive side, we show that, when fixing some constant minimum density μ of the sought subgraph, Densestk-Subgraph becomes fixed-parameter tractable with respect to either of the parameters maximum degree of G and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for Densestk-Subgraph with respect to the combined parameter “degeneracy of G and |V|−k”.On the negative side, we find that Densestk-Subgraph is W[1]-hard with respect to the combined parameter “solution size k and degeneracy of G”. We furthermore strengthen a previous hardness result for Densestk-Subgraph (Cai, 2008) by showing that for every fixed μ, 0<μ<1, the problem of deciding whether G contains a subgraph of density at least μ is W[1]-hard with respect to the parameter |V|−k.Our positive results are obtained by an algorithmic framework that can be applied to a wide range of Fixed-Cardinality Optimization problems.

A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turán's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)−t edges contains a p-partite subgraph with at least e(Tn,p)−2t edges.As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemerédi's regularity lemma) for the classical stability result of Simonovits [25].

Geometric Biplane Graphs I: Maximal Graphs

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.

On Extremal Graphs with at Most $$\ell $$ ℓ Internally Disjoint Steiner Trees Connecting Any $$n-1$$ n - 1 Vertices

The maximum local connectivity was first introduced by Bollobas. The problem of determining the maximum number of edges in a graph with $$\overline{\kappa }\le \ell $$?¯≤l has been studied extensively. We consider a generalization of the above concept and problem. For $$S\subseteq V(G)$$S⊆V(G) and $$|S|\ge 2$$|S|?2, the generalized local connectivity$$\kappa (S)$$?(S) is the maximum number of internally disjoint trees connecting $$S$$S in $$G$$G. The parameter $$\overline{\kappa }_k(G)=\max \{\kappa (S)\,|\,S\subseteq V(G),|S|=k\}$$?¯k(G)=max{?(S)|S⊆V(G),|S|=k} is called the maximum generalized local connectivity of $$G$$G. In this paper the problem of determining the largest number $$f(n;\overline{\kappa }_k\le \ell )$$f(n??¯k≤l) of edges for graphs of order $$n$$n that have maximum generalized local connectivity at most $$\ell $$l is considered. The exact value of $$f(n;\overline{\kappa }_k\le \ell )$$f(n??¯k≤l) for $$k=n,n-1$$k=n,n-1 is determined. For a general $$k$$k, we construct a graph to obtain a sharp lower bound.

On k-visibility graphs

We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we bound the maximum thickness of semi-bar k-visibility graphs between ⎡2/3(k + 1) ⎤ and 2k. In addition we show that the maximum number of edges in arc and circle k-visibility graphs on n vertices is at most (k+1)(3n−k−2) for n &gt; 4k+4 and (n 2) for n ≤ 4k+4, while the maximum chromatic number is at most 6k+6. In semi-arc k-visibility graphs on n vertices, we show that the maximum number of edges is (n 2) for n ≤ 3k+3 and at most (k+1)(2n−(k+2)/2) for n &gt; 3k+3, while the maximum chromatic number is at most 4k+4.

A BDDC Algorithm with Enriched Coarse Spaces for Two-Dimensional Elliptic Problems with Oscillatory and High Contrast Coefficients

A balancing domain decomposition by constraints (BDDC) algorithm with enriched coarse spaces is developed and analyzed for two-dimensional elliptic problems with oscillatory and high contrast coefficients. To obtain a robust algorithm based on the classical BDDC framework for conforming finite element methods, a set of enriched primal unknowns is constructed. The enriched component of the primal unknowns is chosen to reflect the local structures of the coefficient by solving two types of generalized eigenvalue problems. These problems are solved locally for every two adjacent subdomains, so that the computations of the enriched primal unknowns are very efficient. Given a tolerance, dominant eigenfunctions with eigenvalues larger than this tolerance are precomputed and used to form coarse basis functions. The resulting condition number by using this enriched coarse space is proved to be bounded above by this tolerance and the maximum number of edges per subdomain, independent of the contrast of the given coefficient. Furthermore, numerical results are presented for various model problems and various choices of primal unknowns to show the performance of the proposed algorithm.

Random linear network coding with probabilistic polynomial-time wiretap adversary

Linear network coding can achieve the network capacity in the single source multicast case. For the single source multicast network with wiretap adversary, Cai and Yeung 2002 suggest an information-secure linear coding which tightly achieves the transmitting rate upper bound C − t, where C is the capacity of the network and t is the maximum number of edges with wiretap adversary. In this paper, we consider the cryptographic security for the messages in the random linear network coding setting. The indistinguishability under chosen-message-attack or CMA-security from probabilistic polynomial-time PPT wiretap adversary is defined. With a pseudorandom generator against linear cryptanalysis, a peer-to-peer protocol is constructed for multicast network using random linear network coding. When the adversary wiretap less than C edges, the protocol is CMA-secure and attains transmitting rate approaching the network's capacity C from below, asymptotically in the expanding ratio i.e., length of output string/length of input string of the pseudorandom generator. Compared with secret-key-based scheme, no secret is needed by the receiver nodes, which means the key managing issues for dynamic networks would never come up.

Chromatic kernel and its applications

In this paper, we study the following Chromatic kernel (CK) problem: given an $$n$$n-partite graph (called a chromatic correlation graph) $$G=(V,E)$$G=(V,E) with $$V=V_{1}\bigcup \cdots \bigcup V_{n}$$V=V1???Vn and each partite set $$V_{i}$$Vi containing a constant number $$\lambda $$? of vertices, compute a subgraph $$G[V_{CK}]$$G[VCK] of $$G$$G with exactly one vertex from each partite set and the maximum number of edges or the maximum total edge weight, if $$G$$G is edge-weighted (among all such subgraphs). CK is a new problem motivated by several applications and no existing algorithm directly solves it. In this paper, we first show that CK is NP-hard even if $$\lambda =2$$?=2, and cannot be approximated within a factor of $$16/17$$16/17 unless P = NP. Then, we present a random-sampling-based PTAS for dense CK. As its application, we show that CK can be used to determine the pattern of chromosome associations in the nucleus for a population of cells. We test our approach by using random and real biological data; experimental results suggest that our approach yields near optimal solutions, and significantly outperforms existing approaches.

Extremal Numbers for Odd Cycles

We describe theC2k+1-free graphs onnvertices with maximum number of edges. The extremal graphs are unique forn∉ {3k− 1, 3k, 4k− 2, 4k− 1}. The value ofex(n,C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.We obtain that the bound forn0(C2k+1) is 4kin the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

A counterexample to sparse removal

The Turán number of a graph H, denoted ex(n,H), is the maximum number of edges in an n-vertex graph with no subgraph isomorphic to H. Solymosi (2011) conjectured that if H is any graph and ex(n,H)=O(nα) where α>1, then any n-vertex graph with the property that each edge lies in exactly one copy of H has o(nα) edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when H is a non-bipartite graph, in particular when H is a triangle, due to Ruzsa and Szemerédi (1978). Using Sidon sets we exhibit infinitely many bipartite graphs H for which the conjecture is false.

Turán problems and shadows I: Paths and cycles

A k-path is a hypergraph Pk={e1,e2,…,ek} such that |ei∩ej|=1 if |j−i|=1 and ei∩ej=∅ otherwise. A k-cycle is a hypergraph Ck={e1,e2,…,ek} obtained from a (k−1)-path {e1,e2,…,ek−1} by adding an edge ek that shares one vertex with e1, another vertex with ek−1 and is disjoint from the other edges.Let exr(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We prove that for fixed r≥3, k≥4 and (k,r)≠(4,3), for large enough n:exr(n,Pk)=exr(n,Ck)=(nr)−(n−⌊k−12⌋r)+{0if k is odd(n−⌊k−12⌋−2r−2)if k is even and we characterize all the extremal r-graphs. We also solve the case (k,r)=(4,3), which needs a special treatment. The case k=3 was settled by Frankl and Füredi.This work is the next step in a long line of research beginning with conjectures of Erdős and Sós from the early 1970s. In particular, we extend the work (and settle a conjecture) of Füredi, Jiang and Seiver who solved this problem for Pk when r≥4 and of Füredi and Jiang who solved it for Ck when r≥5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.

An Extension of Turán's Theorem, Uniqueness and Stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turán graph turns out to be the unique extremal graph. For $(r-1)|M|&lt;n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.