Minimum Possible Number Of Edges Research Articles

Orthogonal Dissection into Few Rectangles

AbstractWe describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.

О единственности минимального рёберного 1-расширения гиперкуба Q4

One of the important properties of reliable computing systems is their fault tolerance. To study fault tolerance, you can use the apparatus of graph theory. Minimal edge extensions of a graph are considered, which are a model for studying the failure of links in a computing system. A graph G* = (V*,α*) with n vertices is called a minimal k-edge extension of an n-vertex graph G = (V, α) if the graph G is embedded in every graph obtained from G* by deleting any of its k edges and has the minimum possible number of edges. The hypercube Qn is a regular 2n-vertex graph of order n, which is the Cartesian product of n complete 2-vertex graphs K2. The hypercube is a common topology for building computing systems. Previously, a family of graphs Q*n was described, whose representatives for n>1 are minimal edge 1-extensions of the corresponding hypercubes. In this paper, we obtain an analytical proof of the uniqueness of minimal edge 1-extensions of hypercubes for n≤4 and establish a general property of an arbitrary minimal edge 1-extension of a hypercube Qn for n>2: it does not contain edges connecting vertices, the distance between which in the hypercube is equal to 2.

Colorings of hypergraphs with large number of colors

The paper deals with the well-known problem of Erdős and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let m(n,r) denote the minimum possible number of edges in an n-uniform non-r-colorable hypergraph. We show that for r>n, c1nlnn⩽m(n,r)rn⩽C1n3lnn, where c1,C1>0 are some absolute constants. Moreover, we obtain similar bounds for d(n,r), which is equal to the minimum possible value of the maximum edge degree in an n-uniform non-r-colorable hypergraph. If r>n, then c2nlnn⩽d(n,r)rn−1⩽C2n3lnn, where c2,C2>0 are some other absolute constants.

Colorings of hypergraphs with large number of colors

The paper deals with the well-known problem of Erdős and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let m(n,r) denote the minimum possible number of edges in an n-uniform non-r-colorable hypergraph. We show that for r<n,c1nln⁡n≤m(n,r)rn≤C1n3ln⁡n, where c1, C1>0 are some absolute constants.

Index Coding Capacity: How Far Can One Go With Only Shannon Inequalities?

An interference alignment perspective is used to identify the simplest instances (minimum possible number of edges in the alignment graph, no more than 2 interfering messages at any destination) of index coding problems where non-Shannon information inequalities are necessary for capacity characterization. In particular, this includes the first known example of a multiple unicast (one destination per message) index coding problem where non-Shannon information inequalities are shown to be necessary. The simplest multiple unicast example has 7 edges in the alignment graph and 11 messages. The simplest multiple groupcast (multiple destinations per message) example has 6 edges in the alignment graph, 6 messages, and 10 receivers. For both the simplest multiple unicast and multiple groupcast instances, the best outer bound based on only Shannon inequalities is $\frac{2}{5}$, which is tightened to $\frac{11}{28}$ by the use of the Zhang-Yeung non-Shannon type information inequality, and the linear capacity is shown to be $\frac{5}{13}$ using the Ingleton inequality. Conversely, identifying the minimal challenging aspects of the index coding problem allows an expansion of the class of solved index coding problems up to (but not including) these instances.

Diametral broadcast graphs

This paper studies the family of graphs with broadcast time equal to their diameter. The diametral broadcast graph (dbg) problem is to answer the question whether for a given n and d a graph on n vertices can be constructed whose diameter and broadcast time are equal to d. This paper presents several dbg constructions. Together, they solve the dbg problem for all the possible combinations of values of n and d. We also define the diametral broadcast function DB(n,d) as the minimum possible number of edges in a dbg on n vertices and diameter d. We describe all the trees on n vertices with diametral broadcast time. These trees give the exact value for DB(n,d) when tree based dbg construction is possible. For the general case we give an upper bound on DB(n,d).

Two notions of unit distance graphs

A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph.In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed d the number of faithful distance graphs in Rd on n labelled vertices is 2(1+o(1))dnlog2n, and give a short proof of the known fact that the number of distance graphs in Rd on n labelled vertices is 2(1−1/⌊d/2⌋+o(1))n2/2. We also study the behavior of several Ramsey-type quantities involving these graphs.In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in Rd.

Analytical characterizations of some classes of optimal strongly attack-tolerant networks and their Laplacian spectra

This paper analytically characterizes certain classes of low-diameter strongly attack-tolerant networks of arbitrary size, which are globally optimal in the sense that they contain the minimum possible number of edges. Strong attack tolerance property of level $$R$$ R implies that a network preserves connectivity and diameter after the deletion of up to $$R-1$$ R - 1 network elements (vertices and/or edges). In addition to identifying such optimal network configurations, we explicitly derive their entire Laplacian spectra, that is, all eigenvalues and eigenvectors of the graph Laplacian matrix. Each of these eigenvalues is by itself a solution to a global optimization problem; thus, the results of this study show that these optimization problems yield analytical solutions for the considered classes of networks. As an important special case, we show that the algebraic connectivity (i.e., the second-smallest eigenvalue of the Laplacian) considered as a function on all networks with fixed vertex connectivity $$R$$ R reaches its maximum on the optimal $$R$$ R -robust 2-club, which has diameter 2 and strong attack tolerance of level $$R$$ R . We also demonstrate that the obtained results have direct implications on the exact calculation of convergence speed of consensus algorithms utilizing the entire Laplacian spectrum, which is in contrast to traditionally used simulation-based estimates through just the algebraic connectivity.

The Complexity of Obtaining a Distance-Balanced Graph

An unweighted, connected graph is distance-balanced (also called self-median) if there exists a number $d$ such that, for any vertex $v$, the sum of the distances from $v$ to all other vertices is $d$. An unweighted connected graph is strongly distance-balanced (also called distance-degree regular) if there exist numbers $d_1,d_2,d_3,\dots$ such that, for any vertex $v$, there are precisely $d_k$ vertices at distance $k$ from $v$. We consider the following optimization problem: given a graph, add the minimum possible number of edges to obtain a (strongly) distance-balanced graph. We show that the problem is NP-hard for graphs of diameter three, thus answering the question posed by Jerebic et al. [Distance-balanced graphs; Ann. Comb. 2008]. In contrast, we show that the problem can be solved in polynomial time for graphs of diameter 2.

Ranking Tournaments

A tournament is an oriented complete graph. The feedback arc set problem for tournaments is the optimization problem of determining the minimum possible number of edges of a given input tournament T whose reversal makes T acyclic. Ailon, Charikar, and Newman showed that this problem is NP-hard under randomized reductions. Here we show that it is in fact NP-hard. This settles a conjecture of Bang-Jensen and Thomassen.

The size of minimum 3‐trees: Cases 3 and 4 mod 6

A 3-uniform hypergraph is called a minimum 3-tree, if for any 3-coloring of its vertex set there is a heterochromatic edge and the hypergraph has the minimum possible number of edges. Here we show that the number of edges in such 3-tree is for any number of vertices n ≡ 3, 4 (mod 6). © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 157-166, 1999

The size of minimum 3-trees: Cases 3 and 4 mod 6

A 3-uniform hypergraph is called a minimum 3-tree, if for any 3-coloring of its vertex set there is a heterochromatic edge and the hypergraph has the minimum possible number of edges. Here we show that the number of edges in such 3-tree is for any number of vertices n ≡ 3, 4 (mod 6). © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 157-166, 1999

Superconcentrators of depths 2 and 3; odd levels help (rarely)

It is shown that the minimum possible number of edges in an n-superconcentrator of depth 3 is Θ(n log log n), whereas the minimum possible number of edges in an n-superconcentrator of depth 2 is Ω(n(log n)3/2) (and is O(n(log n)2)).

Tighter time bounds on fault‐tolerant broadcasting and gossiping

AbstractConsider a network in which n processors are connected by unreliable lines and are allowed to communicate with at most one other processor at a time. In this paper, the problems of broadcasting and gossiping are considered. Broadcast is the task of transmitting a message, originated at one node, to all other nodes in the network. Gossiping refers to the process of information dissemination when each processor knows a unique item of information and must transmit it to all the other processors in the network. In this paper, new bounds on fault‐tolerant broadcasting and gossiping times are obtained. The given bounds improve on previously known results. In particular, if n is a power of two, the given broadcast and gossiping schemes require minimum time and are supported by a network having the minimum possible number of edges.

How well can a graph be n-colored?

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω( epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph ( e⩾ p⩾ n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K m in K p and equicolored graphs in K p.

The minimum number of edges in graphs with prescribed paths

LetM be anm-by-n matrix with entries in {0,1,⋯,K}. LetC(M) denote the minimum possible number of edges in a directed graph in which (1) there arem distinguished vertices calledinputs, andn other distinguished vertices calledoutputs; (2) there is no directed path from an input to another input, from an output to another output, or from an output to an input; and (3) for all 1 ≤i ≤m and 1 ≤j ≤n, the number of directed paths from thei-th input to thej-th output is equal to the (i,j)-th entry ofM. LetC(m,n,K) denote the maximum ofC(M) over allm-by-n matricesM with entries in {0,1,⋯,K}. We assume (without loss of generality) thatm ≥n, and show that ifm=(K+1) 0(n) andK=22 0(m) , thenC(m,n,K)= H/logH + 0(H/logH), whereH=mnlog(K + 1) and all logarithms have base 2. The proof involves an interesting problem of Diophantine approximation, which is solved by means of an unusual continued fraction expansion.

Extremal graphs of diameter 4

The object of this paper is to determine the minimum possible number of edges for a graph of order n and diameter at most four having the property that by suppressing any vertex the subgraph obtained has diameter at most λ ≥ 4. Results for λ ≥ 6 will be proved.