For each positive integer n, we construct a Steiner triple system of order v = 2(3n) + 1 with no almost parallel class; that is, with no set of v−1 3 disjoint triples. In fact, we construct families of (v, k, λ)-designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to 1 (mod 6) without almost parallel classes were the projective trip...

Let G l and G 2 be (simple) graphs. The Ramsey*number r(Gl ,G 2) is the smallest integer n such that if one colors the complete graph Kn in two colors I and II, then either color I contains G 1 as a subgraph or color II contains G 2. The systematic study of r(Gl ,G2) was initiated by F. Harary, although there were a few previous scattered results of We also note here that notation not defined f...

A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based indices such as Wiener index, hyper-Wiener Harary Tratch – Stankevitch Zefirov index. In present study, we determine closed form for third type chain hex-derived network dime...

In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-M...

In the buy-at-bulk k-Steiner tree (or rent-or-buy k-Steiner tree) problem we are given a graph G(V,E) with a set of terminals T ⊆ V including a particular vertex s called the root, and an integer k ≤ |T |. There are two cost functions on the edges of G, a buy cost b : E −→ R+ and a rent cost r : E −→ R+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑

In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1, 2, . . . ,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97 ·n · logW ) using O(n · lognW · log k) space. This is the first single-exponential time, polynomial-sp...

Given a set of i (i = 1; 2; : : :; k) orientations (angles) in the plane, one can deene a distance function which induces a metric in the plane, called the orientation metric 3]. In the special case where all the angles are equal, we call the metric a uniform orientation metric 2]. Speciically, if there are orientations, forming angles ii ; 0 i ?1, with the x-axis, where 2 is an integer, we cal...

We consider the currently strongest Steiner tree approximation algorithm that has recently been published by Goemans, Olver, Rothvoß and Zenklusen (2012). It first solves a hypergraphic LP relaxation and then applies matroid theory to obtain an integral solution. The cost of the resulting Steiner tree is at most (1.39 + ε)-times the cost of an optimal Steiner tree where ε tends to zero as some ...