Tzvetalin S. Vassilev

Algorithms for optimal area triangulations of a convex polygon

Given a convex polygon with n vertices in the plane, we are interested in triangulations of its interior, i.e., maximal sets of non-intersecting diagonals that subdivide the interior of the polygon into triangles. The MaxMin area triangulation is the triangulation of the polygon that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. We present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in O(n^2logn) time and O(n^2) space. The algorithms use dynamic programming and a number of geometric properties that are established within the paper.

The relative neighbourhood graph is a part of every 30°-triangulation

We study sets of points in the two-dimensional Euclidean plane. The relative neighbourhood graph (RNG) of a point set is a straight line graph that connects two points from the point set if and only if there is no other point in the set that is closer to both points than they are to each other. A triangulation of a point set is a maximal set of nonintersecting line segments (called edges) with vertices in the point set. We introduce angular restrictions in the triangulations. Using the well-known method of exclusion regions, we show that the relative neighbourhood graph is a part of every triangulation all of the angles of which are greater than or equal to 30^o.

Solution of an outstanding conjecture: the non-existence of universal cycles with k=n−2

Abstract A universal cycle for k -subsets of an n -set, {1,2,…, n }, is a cyclic sequence of ( n k ) integers with the property that each subset of {1,2,…, n } of size k appears exactly once consecutively in the sequence. This problem was first posed by Chung et al. (Discrete Math. 110 (1992) 43) and solved for k =2,3,4,6 by Jackson and Hurlbert (Ph.D. Thesis, Rutgers University, New Brunswick, NJ, 1990; SIAM J. Discrete Math. 7(4) (1994) 598; Discrete Math. 137 (1995) 241; Personal communication, 1999). Both Jackson and Hurlbert noted the difficulty of finding universal cycles with k ⩾⌈ n /2⌉. Jackson has found some of these but conjectured that universal cycles never exist when k = n −2. We prove this result and give some bounds on the longest word not repeating any ( n −2)-subset and also the shortest word that contains all at least once.

Picture Maze Generation by Successive Insertion of Path Segment

Picture maze is a maze on which some picture appears when it is solved. This paper aims to present a very simple method to construct a picture maze. It is always possible to construct a spanning tree on a connected region of a binary picture. A spanning tree connects all vertices on the region. A 2-by-2 extended picture of a connected picture is the one such that each cell in the picture is transformed to four cells which constitute a square block with the cells. Then by analogy, it is always possible to construct a block spanning tree on the extended picture region, where a block is the square unit of four cells. Construction of the block spanning tree and the generation of the corresponding Hamiltonian path can be done at the same time. The Hamiltonian path is constructed along a data structure of a linked list in a simple manner. It traverses every cell on the 2-by-2 extended picture. The similar procedure but with the smaller block with two vertices, instead of four, can be applied to a non-extended picture, an ordinary one, which produces a near Hamiltonian path, which can then be a maze solution path. The near Hamiltonian path traverses nearly all the vertices on the picture, hence depicting a picture on the maze when it is solved. This paper demonstrates the method and its efficiency.

Overview of Graph Colouring and some Ramsey-type Numbers

We introduce the concept of graph colouring and discuss some classical results in this area. In particular, we consider the problem of finding the minimal graphs, complete or not, whose vertex or edge colouring contains or avoids certain subgraphs. This is generally known as Ramsey theory. We give short proofs of some elementary results in this area, and discuss their relationship to colouring integer sequences.

Math Circles in North Bay – the Northern Experience

Identifying mathematically talented children and working with them in various forms to help them achieve their full potential has been a priority in many countries around the world in the last 50 years [1-3]. In the last 20 years, special attention has been paid to this in North America as well [4-6]. One of the most common forms is the so-called Mathematical Circles. In recent years, a large number of universities have taken leadership in organizing Mathematical Circles in their respective cities and communities. Many of these activities are related to mathematical Olympiads or other competitions [7,8]. They also help prepare children for the transition into college/university at an advanced level. There are several forums for people involved in this activity, including the Special Interest Group on Mathematical Circles of the Mathematical Association of America [9-11]. In this paper, we describe the experience of a group of faculty and students at Nipissing University in running the Math Circles in North Bay, Ontario. *Corresponding author: E-mail: tzvetalv@nipissingu.ca Huntington & Vassilev; BJESBS, 6(3), 189-195, 2015; Article no.BJESBS.2015.055