Number Of Complete Graph Research Articles

Integer k-matching preclusion of graphs

As a generalization of matching preclusion number of a graph, we provide the (strong) integer $k$-matching preclusion number, abbreviated as $MP^{k}$ number ($SMP^{k}$ number), which is the minimum number of edges (vertices and edges) whose deletion results in a graph that has neither perfect integer $k$-matching nor almost perfect integer $k$-matching. In this paper, we obtain a necessary condition of graphs with an almost-perfect integer $k$-matching and a relational expression between the matching number and the integer $k$-matching number of bipartite graphs. And then the $MP^{k}$ number and the $SMP^{k}$ number of complete graphs, bipartite graphs and arrangement graphs are obtained, respectively.

Slope Number on Complete Graphs

Objective: A straight line drawing is a mapping of an edge into a straight line segment. The minimum number of distinct slopes used in a straight line drawing of a graph G is called the slope number of the graph G. In this paper the slope number of complete graph is studied elaborately. Methods: This optimization problem is NP-Hard for any arbitrary graph. A canonical way of drawing of a complete graph is an existing one. In present paper, we consider the edges of a complete graph are straight line segments in order to obtain the number of slopes. Findings: This paper interprets the characterization of slopes in complete graph according to an odd and even number of edges and investigated in detail. Moreover, the slope number of a complete graph is compared with the chromatic number of complete graph and the results are observed. Applications/Improvement: Slope number is one of the quality measures of graph drawing. It is used to find out different layout methods for the same graph.

Discovery of Influenza A Virus Sequence Pairs and Their Combinations for Simultaneous Heterosubtypic Targeting that Hedge against Antiviral Resistance.

The multiple circulating human influenza A virus subtypes coupled with the perpetual genomic mutations and segment reassortment events challenge the development of effective therapeutics. The capacity to drug most RNAs motivates the investigation on viral RNA targets. 123,060 segment sequences from 35,938 strains of the most prevalent subtypes also infecting humans–H1N1, 2009 pandemic H1N1, H3N2, H5N1 and H7N9, were used to identify 1,183 conserved RNA target sequences (≥15-mer) in the internal segments. 100% theoretical coverage in simultaneous heterosubtypic targeting is achieved by pairing specific sequences from the same segment (“Duals”) or from two segments (“Doubles”); 1,662 Duals and 28,463 Doubles identified. By combining specific Duals and/or Doubles to form a target graph wherein an edge connecting two vertices (target sequences) represents a Dual or Double, it is possible to hedge against antiviral resistance besides maintaining 100% heterosubtypic coverage. To evaluate the hedging potential, we define the hedge-factor as the minimum number of resistant target sequences that will render the graph to become resistant i.e. eliminate all the edges therein; a target sequence or a graph is considered resistant when it cannot achieve 100% heterosubtypic coverage. In an n-vertices graph (n ≥ 3), the hedge-factor is maximal (= n– 1) when it is a complete graph i.e. every distinct pair in a graph is either a Dual or Double. Computational analyses uncover an extensive number of complete graphs of different sizes. Monte Carlo simulations show that the mutation counts and time elapsed for a target graph to become resistant increase with the hedge-factor. Incidentally, target sequences which were reported to reduce virus titre in experiments are included in our target graphs. The identity of target sequence pairs for heterosubtypic targeting and their combinations for hedging antiviral resistance are useful toolkits to construct target graphs for different therapeutic objectives.

Quasi‐randomness of graph balanced cut properties

AbstractQuasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α1,…,αk be positive reals satisfying \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document} \begin{align*}\sum_{i} \alpha_i = 1\end{align*} \end{document} and (α1,…,αk)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V 1,…,V k of size α1n,…,αkn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Bounding the pseudoachromatic index of the complete graph via projective planes

Abstract Let q = 2 β and n = q 2 + q + 1 . Further, let G = L ( K n ) be the complete line graph and ψ ( G ) its pseudoachromatic number. By exhibiting an explicit colouring of E ( K n ) , we show that ψ ( G ) ⩾ q 3 + q . This result improves the bound ψ ( G ) ⩾ q 3 + 1 due to Jamison (1989) [Jamison, R.E.; On the edge achromatic numbers of complete graphs, Discrete. Math. 74 (1989) 99–115].

Hierarchical management protocol for constructing a QoS communication path in wireless Ad Hoc networks

An Ad Hoc network consists of mobile hosts that can dynamically construct a wireless network without base stations. Due to the limited communication range, a source host usually needs other hosts to relay messages to the destination in a multi-hop manner. Consequently, establishing a routing path from the source to the destination is a basic requirement for providing communication service between any pair of mobile hosts. This study proposes a two-level management approach for efficiently constructing and maintaining a QoS routing path in Ad Hoc wireless networks, significantly reducing the quantity of control packets. In the first phase, the mobile hosts are partitioned into a number of complete graphs, each represented by a Supernode managed by an agent. The Ad Hoc network topology is thus transformed to an Agent-based Graph ( AG). In the second phase, some agents of a larger degree than neighboring agents are selected as core nodes. The core nodes then virtually construct a Core Graph ( CG). The proposed two-level hierarchical management and bandwidth-looking-ahead technologies can efficiently establish and maintain a QoS communication path at a low control packet cost. Simulation results indicate that the proposed management model significantly reduces the number of control packets in areas with very large numbers of mobile hosts.

Enumerations of the maximum rectilinear crossing numbers of complete and complete multi-partite graphs

The maximum rectilinear crossing number of a complete graph of order n is known to be . In this paper, the edge-set of a complete graph is partitioned into L layers, where the contribution of each layer on the maximum rectilinear crossing number is particularly described. We obtain combinatorial identities and triangular arrays that can be used to calculate . The results obtained from the enumeration of complete graphs are then used to calculate the maximum rectilinear crossing number of the complete multi-partite graphs.

A Probabilistic Counting Lemma for Complete Graphs

We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.

On the galactic number of a hypercube

A galaxy is a union of vertex disjoint stars. The galactic number of a graph is the minimum number of galaxies which partition the edge set. The galactic number of complete graphs is determined. This result is used to give bounds on the galactic number of binary cube graphs. The problem of determining the galactic number of a graph is shown to be NP-complete.