scholarly journals The super-connectivity of odd graphs and of their Kronecker double cover

2020 ◽  
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
John Baptist Gauci
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Kronecker Product ◽  
Double Cover ◽  
Kneser Graph ◽  
Classical Notion ◽  
Super Connectivity ◽  
Minimum Number ◽  
Kneser Graphs

The study of connectivity parameters forms an integral part of the research conducted in establishing the fault tolerance of networks. A number of variations on the classical notion of connectivity have been proposed and studied. In particular, the super--connectivity asks for the minimum number of vertices that need to be deleted from a graph in order to disconnect the graph without creating isolated vertices. In this work, we determine this value for two closely related families of graphs which are considered as good models for networks, namely the odd graphs and their Kronecker double cover. The odd graphs are constructed by taking all possible subsets of size $k$ from the set of integers $\{1,\ldots,2k+1\}$ as vertices, and defining two vertices to be adjacent if the corresponding $k$-subsets are disjoint; these correspond to the Kneser graphs $KG(2k+1,k)$. The Kronecker double cover of a graph $G$ is formed by taking the Kronecker product of $G$ with the complete graph on two vertices; in the case when $G$ is $KG(2k+1,k)$, the Kronecker double cover is the bipartite Kneser graph $H(2k+1,k)$. We show that in both instances, the super--connectivity is equal to $2k$.

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

The Restricted Edge-Connectivity of Kronecker Product Graphs

2019 ◽  
Vol 29 (03) ◽  
pp. 1950012
Author(s):  
Tianlong Ma ◽  
Jinling Wang ◽  
Mingzu Zhang
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Kronecker Product ◽  
Sufficient Condition ◽  
Connected Component ◽  
Edge Connectivity ◽  
Product Graphs ◽  
Minimum Number ◽  
Vertex Set ◽  
Product Of Graphs

The restricted edge-connectivity of a connected graph [Formula: see text], denoted by [Formula: see text], if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, it is proved that [Formula: see text] for any graph [Formula: see text] and a complete graph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is minimum edge-degree of [Formula: see text], and a sufficient condition such that [Formula: see text] is [Formula: see text]-optimal is acquired.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

Reliability Analysis of the Generalized Exchanged Hypercube

2020 ◽  
Vol 30 (02) ◽  
pp. 2050009
Author(s):  
Qifan Zhang ◽  
Liqiong Xu ◽  
Weihua Yang ◽  
Shanshan Yin
Keyword(s):  
Fault Tolerance ◽  
Reliability Analysis ◽  
Network Structure ◽  
Complete Graph ◽  
Interconnection Network ◽  
Natural Extension ◽  
Graph Model ◽  
Text Component ◽  
Optimal Formula

Let [Formula: see text] be a non-complete graph, a subset [Formula: see text] is called a [Formula: see text]-component cut of [Formula: see text], if [Formula: see text] is disconnected and has at least [Formula: see text] components. The cardinality of the minimum [Formula: see text]-component cut is the [Formula: see text]-component connectivity of [Formula: see text] and is denoted by [Formula: see text]. The [Formula: see text]-component connectivity is a natural extension of the classical connectivity. As an application, the [Formula: see text]-component connectivity can be used to evaluate the reliability and fault tolerance of an interconnection network structure based on a graph model. In a previous work, E. Cheng et al. obtained the [Formula: see text]-component connectivity of the generalized exchanged hypercube [Formula: see text] for [Formula: see text] and [Formula: see text]. In this paper, we continue the work and determine that [Formula: see text] for [Formula: see text]. Moreover, we show that every optimal [Formula: see text]-component cut of [Formula: see text] is trivial for [Formula: see text] and [Formula: see text].

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals The super-connectivity of Kneser graphs

2019 ◽  
Vol 39 (1) ◽  
pp. 5 ◽  
Author(s):  
Gülnaz Boruzanli Ekinci ◽  
John Baptist Gauci
Keyword(s):  
Super Connectivity ◽  
Kneser Graphs

scholarly journals On the $P_3$-Hull Number of Kneser Graphs

10.37236/9903 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Luciano N. Grippo ◽  
Adrián Pastine ◽  
Pablo Torres ◽  
Mario Valencia-Pabon ◽  
Juan C. Vera
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Lower And Upper Bounds ◽  
Kneser Graph ◽  
Graph Homomorphisms ◽  
The Family ◽  
Vertex Set ◽  
Kneser Graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph. In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.

scholarly journals Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Martin Balko ◽  
Josef Cibulka ◽  
Karel Král ◽  
Jan Kynčl
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Positive Answer ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Ordered Graphs ◽  
Constant Interval ◽  
Monochromatic Copy ◽  
Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

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