Balanced semi-Latin rectangles : properties, existence and constructions for block size two

There exists a set of designs which form a subclass of semi-Latin rectangles. These designs, besides being semi-Latin rectangles, exhibit an additional property of balance; where no two distinct pairs of symbols (treatments) differ in their concurrences, that is, each pair of distinct treatments concur a constant number of times in the design. Such a design exists for a limited set of parameter combinations. We designate it a balanced semi-Latin rectangle (BSLR) and give some properties, and necessary conditions for its existence. Furthermore, algorithms for constructing the design for experimental situations where there are two treatments in each row-column intersection (block) are also given.Publisher PDFPeer reviewe

Zarankiewicz numbers near the triple system threshold

For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$. Here, we extend this by determining $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3$ and, when $m$ is large, for all $n \geq \binom{m}{2}/6+O(m)$.Comment: 18 pages, 4 figure

UNIFORM THREE-CLASS REGULAR PARTIAL STEINER TRIPLE SYSTEMS WITH UNIFORM DEGREES

A Partial Steiner Triple system (X, T) is a finite set of points C and a collection T of 3-element subsets of C that every pair of points intersect in at most 1 triple. A 3-class regular PSTS (3-PSTS) is a PSTS where the points can be partitioned into 3 classes (each class having size m, n and p respectively) such that no triple belongs to any class and any two points from the same class occur in the same number of triples (a, b and c respectively). The 3-PSTS is said to be uniform if m = n = p. In this thesis, we have mostly focused on the existence of uniform 3-PSTS with uniform degrees (a = b = c)

Spectrum of Sizes for Perfect Deletion-Correcting Codes

One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the Levenshte\u{\i}n distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect $t$-deletion-correcting code, given the length $n$ and the alphabet size~$q$. In this paper, we determine completely the spectrum of possible sizes for perfect $q$-ary 1-deletion-correcting codes of length three for all $q$, and perfect $q$-ary 2-deletion-correcting codes of length four for almost all $q$, leaving only a small finite number of cases in doubt.Comment: 23 page

On nesting of G-decompositions of λKv where G has four nonisolated vertices or less

AbstractThe complete multigraph λKv is said to have a G-decomposition if it is the union of edge disjoint subgraphs of Kv each of them isomorphic to a fixed graph G. The spectrum problem for G-decompositions of λKv that have a nesting was first considered in the case G=K3 by Colbourn and Colbourn (Ars Combin. 16 (1983) 27–34) and Stinson (Graphs and Combin. 1 (1985) 189–191). For λ=1 and G=Cm (the cycle of length m) this problem was studied in many papers, see Lindner and Rodger (in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory: A Collection of Surveys, Wiley, New York, 1992, p. 325–369), Lindner et al. (Discrete Math. 77 (1989) 191–203), Lindner and Stinson (J. Combin. Math. Combin. Comput. 8 (1990) 147–157) for more details and references. For λ=1 and G=Pk (the path of length k−1) the analogous problem was considered in Milici and Quattrocchi (J. Combin. Math. Combin. Comput. 32 (2000) 115–127). In this paper we solve the spectrum problem of nested G-decompositions of λKv for all the graphs G having four nonisolated vertices or less, leaving eight possible exceptions

Breakout group allocation schedules and the social golfer problem with adjacent group sizes

The current pandemic has led schools and universities to turn to online meeting software solutions such as Zoom and Microsoft Teams. The teaching experience can be enhanced via the use of breakout rooms for small group interaction. Over the course of a class (or over several classes), the class will be allocated to breakout groups multiple times over several rounds. It is desirable to mix the groups as much as possible, the ideal being that no two students appear in the same group in more than one round. In this paper, we discuss how the problem of scheduling balanced allocations of students to sequential breakout rooms directly corresponds to a novel variation of a well-known problem in combinatorics (the social golfer problem), which we call the social golfer problem with adjacent group sizes. We explain how solutions to this problem can be obtained using constructions from combinatorial design theory and how they can be used to obtain good, balanced breakout room allocation schedules. We present our solutions for up to 50 students and introduce an online resource that educators can access to immediately generate suitable allocation schedules