Least Number Of Edges Research Articles

Saturation numbers for Berge cliques

Let F be a graph and H be a hypergraph, both embedded on the same vertex set. We say H is a Berge-F if there exists a bijection ϕ:E(F)→E(H) such that e⊆ϕ(e) for all e∈E(F). We say H is Berge-F-saturated if H does not contain any Berge-F, but adding any missing edge to H creates a copy of a Berge-F. The saturation number satk(n,Berge-F) is the least number of edges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show satk(n,Berge-Kℓ)∼ℓ−2k−1n,for all k,ℓ≥3. Furthermore, we provide some sufficient conditions to imply that satk(n,Berge-F)=O(n) for general graphs F.

Graphs without gap-vertex-labellings: Families and bounds

A gap-[k]-vertex-labelling of a graph G is a pair (π,cπ), where π:V(G)→[k] and cπ:V(G)→{0,1,…,k} is a proper colouring of G such that, for v∈V(G), cπ(v)=maxu∈N(v){π(u)}−minu∈N(v){π(u)}, if d(v)≥2, and cπ(v)=π(u)u∈N(v), otherwise. In this paper, we present an upper bound of O(n2) for the vertex-gap number of arbitrary graphs, which is the least number k such that G admits a gap-[k]-vertex-labelling. We prove that, for n≥4, Kn does not admit any gap-vertex-labelling, regardless of the number of labels. We also characterize the powers of cycles and paths that admit gap-vertex-labellings. Furthermore, we introduce a novel parameter, the gap-strength of a graph, denoted by strgap(G), which is the least number of edges that must be removed from a graph, so it will have a gap-vertex-labelling, and prove that strgap(Kn)∈Ω(n6/5) and strgap(Kn)∈O(n3/2).

Neighbor Connectivity of the Alternating Group Graph

Given a graph [Formula: see text], its neighbor connectivity is the least number of vertices whose deletion along with their neighbors results in a disconnected, complete, or empty graph. The edge neighbor connectivity is the least number of edges whose deletion along with their endpoints results in a disconnected, complete, or empty graph. In this paper, we determine the neighbor connectivity [Formula: see text] and the edge neighbor connectivity [Formula: see text] of the alternating group graph. We show that [Formula: see text], where [Formula: see text] is the [Formula: see text]-dimensional alternating group graph.

2-Colorings of Hypergraphs with Large Girth

A hypergraph $$H=(V,E)$$ has property $$B_k$$ if there exists a 2-coloring of the set $$V$$ such that each edge contains at least $$k$$ vertices of each color. We let $$m_{k,g}(n)$$ and $$m_{k,b}(n)$$ , respectively, denote the least number of edges of an $$n$$ -homogeneous hypergraph without property $$B_k$$ which contains either no cycles of length at least $$g$$ or no two edges intersecting in more than $$b$$ vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for $$m^{*}_k(n)$$ , i.e., for the least number of edges of an $$n$$ -homogeneous simple hypergraph without property $$B_k$$ . Let $$\Delta(H)$$ be the maximal degree of vertices of a hypergraph $$H$$ . By $$\Delta_k(n,g)$$ we denote the minimal degree $$\Delta$$ such that there exists an $$n$$ -homogeneous hypergraph $$H$$ with maximal degree $$\Delta$$ and girth at least $$g$$ but without property $$B_k$$ . In the paper, an upper bound for $$\Delta_k(n,g)$$ is obtained.

A CONSTRUCTION OF GRAPHS WITH POSITIVE RICCI CURVATURE

Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.

Geodesic Gaussian Integer Continued Fractions

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph F, whose vertices are reduced Gaussian rationals in Q∞(i) and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in H3 where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of H3 by Farey octahedrons. A natural distance ϱ on Q∞(i) is also recalled, where ϱ(∞,w)=n is the least number of edges in F from ∞ to w∈Q(i), where n is called the generation of w and a relevant path in F is called a geodesic expansion for w. The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation n−1, n, and n+1. Subsequently, it is seen that there can be at most four Farey neighbors of generation n−1 in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any w is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

Nearly-Regular Hypergraphs and Saturation of Berge Stars

Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is Berge-$G$-saturated if $H$ does not contain a Berge-$G$, but adding any edge to $H$ creates a Berge-$G$. The saturation number for Berge-$G$, denoted $\mathrm{sat}_k(n,\text{Berge-}G)$ is the least number of edges in a $k$-uniform hypergraph that is Berge-$G$-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular $k$-uniform hypergraphs, or $k$-uniform hypergraphs in which every vertex has degree $r$ or $r-1$ for some $r\in \mathbb{Z}$, and less than $k$ vertices have degree $r-1$.

On the Crossing Number of 2-Page Book Drawings of $$K_{n}$$ with Prescribed Number of Edges in Each Page

We consider the problem of determining the 2-page book crossing number of the complete graph $$ K_n $$ when the number of edges in each page is given. We find upper and lower bounds of the right order of magnitude depending on the number of edges in the page with the least number of edges.

Reliable Flexibility Design of Supply Chains Via Extended Probabilistic Expanders

It is well‐known that adding a little flexibility to the right place is an effective strategy to improve the performance of operations in the face of demand uncertainties, to ensure high level of capacity utilization. However, given that system disruptions are ubiquitous, the legacy flexibility designs may perform poorly under disruptions to supply or capacity installations. In this study, we focus on the design of reliable and sparse flexibility structures that consistently meet a reasonable performance criterion under disruptions to both demand and supply. Specifically, we propose a class of structures termed as extended probabilistic expanders, based on the conjecture that the expansion property, rather than the global connectivity, is critical to good performance of the structures. We prove that for a system with n retailers, essentially only O( n) supply routes between suppliers and retailers are necessary to ensure good performance under disruption. In addition, we present an efficient randomized algorithm to construct extended probabilistic expanders, and demonstrate that the construction yields very good structure with the least number of edges asymptotically. We also investigate an extension to systems with structural constraints. Numerical results demonstrate that our design has not only a wide range of applications, but also better performance than a variety of known structures.

Convexity in scientific collaboration networks

Convexity in a network (graph) has been recently defined as a property of each of its subgraphs to include all shortest paths between the nodes of that subgraph. It can be measured on the scale [0, 1] with 1 being assigned to fully convex networks. The largest convex component of a graph that emerges after the removal of the least number of edges is called a convex skeleton. It is basically a tree of cliques, which has been shown to have many interesting features. In this article the notions of convexity and convex skeletons in the context of scientific collaboration networks are discussed. More specifically, we analyze the co-authorship networks of Slovenian researchers in computer science, physics, sociology, mathematics, and economics and extract convex skeletons from them. We then compare these convex skeletons with the residual graphs (remainders) in terms of collaboration frequency distributions by various parameters such as the publication year and type, co-authors' birth year, status, gender, discipline, etc. We also show the top-ranked scientists by four basic centrality measures as calculated on the original networks and their skeletons and conclude that convex skeletons may help detect influential scholars that are hardly identifiable in the original collaboration network. As their inherent feature, convex skeletons retain the properties of collaboration networks. These include high-level structural properties but also the fact that the same authors are highlighted by centrality measures. Moreover, the most important ties and thus the most important collaborations are retained in the skeletons.

Book review

Convexity in a network (graph) has been recently defined as a property of each of its subgraphs to include all shortest paths between the nodes of that subgraph. It can be measured on the scale [0, 1] with 1 being assigned to fully convex networks. The largest convex component of a graph that emerges after the removal of the least number of edges is called a convex skeleton. It is basically a tree of cliques, which has been shown to have many interesting features. In this article the notions of convexity and convex skeletons in the context of scientific collaboration networks are discussed. More specifically, we analyze the co-authorship networks of Slovenian researchers in computer science, physics, sociology, mathematics, and economics and extract convex skeletons from them. We then compare these convex skeletons with the residual graphs (remainders) in terms of collaboration frequency distributions by various parameters such as the publication year and type, co-authors’ birth year, status, gender, discipline, etc. We also show the top-ranked scientists by four basic centrality measures as calculated on the original networks and their skeletons and conclude that convex skeletons may help detect influential scholars that are hardly identifiable in the original collaboration network. As their inherent feature, convex skeletons retain the properties of collaboration networks. These include high-level structural properties but also the fact that the same authors are highlighted by centrality measures. Moreover, the most important ties and thus the most important collaborations are retained in the skeletons.

Length bounds for cycle bases of graphs

The cycle space of a graph G is a vector space over GF(2) which is formed by all Eulerian subgraphs of G with vector addition X⊕Y:=(X∪Y)∖(X∩Y) and scalar multiplication 1⋅X=X, 0⋅X=∅. A base of this vector space is called a cycle base. A cycle base is used to examine the cyclic structure of a graph. The length of a cycle base is the number of its edges in the base. A minimum cycle base is that having the least number of edges. In this paper, we study the length bounds for cycle bases and the minimum cycle bases. A complete characterization is given for a 2-connected graph G with a cycle base of length 2|E(G)|−|V(G)| (it is a lower bound obtained by Leydold and Stadler (1998) [11]). In addition, we derive a sharp lower length bound for minimum cycle bases. As for upper bounds, Horton (1987) showed that the length of a minimum cycle base of a graph with n vertices is at most 3(n−1)(n−2)/2. We improve the bound substantially for graphs on the projective plane where it is at most ⌊13n/2⌋−9.

Convex skeletons of complex networks.

A convex network can be defined as a network such that every connected induced subgraph includes all the shortest paths between its nodes. A fully convex network would therefore be a collection of cliques stitched together in a tree. In this paper, we study the largest high-convexity part of empirical networks obtained by removing the least number of edges, which we call a convex skeleton. A convex skeleton is a generalization of a network spanning tree in which each edge can be replaced by a clique of arbitrary size. We present different approaches for extracting convex skeletons and apply them to social collaboration and protein interactions networks, autonomous systems graphs and food webs. We show that the extracted convex skeletons retain the degree distribution, clustering, connectivity, distances, node position and also community structure, while making the shortest paths between the nodes largely unique. Moreover, in the Slovenian computer scientists coauthorship network, a convex skeleton retains the strongest ties between the authors, differently from a spanning tree or high-betweenness backbone and high-salience skeleton. A convex skeleton thus represents a simple definition of a network backbone with applications in coauthorship and other social collaboration networks.

Weak oddness as an approximation of oddness and resistance in cubic graphs

We consider the question whether the least number ω of odd components in a 2-factor is the same as the least number ωw of odd components in an even factor of a cubic graph. Both invariants appear in the literature under the same name “oddness”. We show that these two invariants are different by constructing graphs G satisfying ρ(G)<ωw(G)<ω(G) (where ρ denotes resistance, i.e. the least number of edges that have to be removed from a cubic graph in order to turn it into a 3-edge-colourable graph). In addition, we demonstrate that the difference between any two of those three invariants may be arbitrarily large. Our results imply that if we replace a vertex of a cubic graph with a triangle, then its oddness (ω) may significantly decrease.

The vertex size-Ramsey number

In this paper, we study an analogue of size-Ramsey numbers for vertex colorings. For a given number of colors r and a graph G the vertex size-Ramsey number of G, denoted by Rˆv(G,r), is the least number of edges in a graph H with the property that any r-coloring of the vertices of H yields a monochromatic copy of G. We observe that Ωr(Δn)=Rˆv(G,r)=Or(n2) for any G of order n and maximum degree Δ, and prove that for some graphs these bounds are tight. On the other hand, we show that even 3-regular graphs can have nonlinear vertex size-Ramsey numbers. Finally, we prove that Rˆv(T,r)=Or(Δ2n) for any tree of order n and maximum degree Δ, which is only off by a factor of Δ from the best possible.

On Simple Multiple Access Networks

We investigate a simple multiple access network (SMAN) where k independent sources of unit rates multicast their information to a set of sinks, via n commonly shared relays. All links are assumed to have unit capacity. Given such a SMAN, a coding scheme for the relays is called optimal if each sink can retrieve all information from the sources under at most ⌊n-k+1/2⌋ node/link errors. We study the problem of designing the sparsest SMAN, i.e., the SMAN that has the least number of edges, that supports an optimal coding scheme for the relays. Additionally, the SMAN must satisfy either of the following constraints: 1) Connection Constraint: Each relay can be connected only to a given subset of sources or 2) Balance Constraint: Each relay must be connected to approximately the same number of sources. We provide two polynomial time algorithms to identify the cases where such a SMAN exists together with its optimal coding scheme designed over sufficiently large fields. One algorithm is based on a nontrivial modification of the well-known Gale-Ryser algorithm, whereas the other is based on a novel generalization of the famous Hall's marriage theorem. We also propose a combinatorial approach to construct optimal coding schemes over small fields and settle the problem for a special case.

Changing the diameter of the locally twisted cube

The hypercube network Q n has been proved to be one of the most popular interconnection networks. The n-dimensional locally twisted cube LTQ n is an important variant of Q n . One of the critical performance factors of an interconnection network is the diameter which determines the maximum communication time between any pair of processors. In this paper, we investigate the diameter variability problems arising from the addition and deletion of edges in LTQ n . We obtain three results in this paper: (1) for any integer n≥2, we find the least number of edges (denoted by ch −(LTQ n )), whose deletion from LTQ n causes the diameter to increase, (2) for any integer n≥2, when ch −(LTQ n ) edges are deleted, the diameter will increase by 1 and (3) for any integer n≥4, the least number of edges whose addition to LTQ n will decrease the diameter is at most 2 n−1.

Maximum Frustration in Bipartite Signed Graphs

A signed graph is a graph where each edge is labeled as either positive or negative. A circle is positive if the product of edge labels is positive. The frustration index is the least number of edges that need to be removed so that every remaining circle is positive. The maximum frustration of a graph is the maximum frustration index over all possible sign labellings. We prove two results about the maximum frustration of a complete bipartite graph $K_{l,r}$, with $l$ left vertices and $r$ right vertices. First, it is bounded above by\[ \frac{lr}{2}\left(1-\frac{1}{2^{l-1}}\binom{l-1}{\lfloor \frac{l-1}{2}\rfloor}\right).\] Second, there is a unique family of signed $K_{l,r}$ that reach this bound. Using this fact, exact formulas for the maximum frustration of $K_{l,r}$ are found for $l \leq 7$.

The realizability problem for Golovach-type functions

The problem mentioned in the title is stated as follows. Consider a function f with some necessary properties of the Golovach function, namely, a piecewise constant nonincreasing right continuous function defined on the set of nonnegative real numbers and taking integer values such that this function is identically equal to 1 at sufficiently large argument values. The problem of realizing the function f in a class \(\mathbb{G}\) of topological graphs is to find a graph \(G \in \mathbb{G}\) such that its Golovach functions coincides with f. Examples of realization of some functions possessing the properties mentioned above are considered. In the simplest case, all graphs for which the function can be realized are described. For less trivial examples, realizability criteria for functions with the properties of the Golovach function in the class of trees and in the class of trees with given edge search number which have the least number of edges are presented.

An Improved Ant-Based Algorithm for Minimum Degree Spanning Tree Problems

A spanning tree of a connected graph is a sub graph, with least number of edges that still spans. The problem of finding degree constraint spanning tree is known to be NP-hard. In this paper we discuss an Ant-Based algorithm for finding minimum degree spanning trees and give improvement of the algorithm. We also show comparisons among the three algorithms and find the best improved Ant-Based algorithm. Extensive experimental results show that our improved algorithm performs very well against other algorithms on a set of 50 problem instances. Keywords - Ant algorithm, Graph algorithms, Heuristic methods, minimum degree spanning tree, I. INTRODUCTION This paper describes Ant-Based algorithms for minimum degree spanning tree (AB-MDST) of unweighted connected graph. This is an interesting, real-world problem that seems well suited to an ant algorithm approach. The AB-MDST problem entails finding a spanning tree such that the maximum degree of a vertex in the tree becomes minimum (2). This concept is useful in the design of telecommunication networks, design of networks for computer communication, design of integrated circuits, energy networks, transportation, logistics, and sewage networks (3). For instance, switches in an actual communication network will have limited number of connections available. Transportation systems must place a limit on the number of roads meeting in one place. The problem of finding degree constraint spanning tree is NP-hard (9). Therefore, heuristics are often used to find good solutions in a reasonable amount of time (10). We have used one type of heuristic called Ant Colony Optimization (ACO) (10). Here, artificial ants move based on local information and pheromone levels. Our algorithm uses cumulative pheromone levels to determine candidate set of edges from which minimum degree spanning trees are built (7). In this paper, we compare 3 algorithms - AB-MDST without local search and without degree constraint, AB-MDST with local search but without degree constraint and the last one AB- MDST with local search and with degree constraint. Extensive experimental results show that AB-MDST with local search and with degree constraint performs very well against other algorithms. The rest of the paper is organized as follow. In section 2 Our Ant-Based algorithm and two improved versions of that algorithm are described. Section 3 compares the performances of the three algorithms. The conclusion is given in section 4.