HOPs and COPs: Room frames with partitionable transversals

In this paper, we construct Room frames with partitionable transversals. Direct and recursive constructions are used to find sets of disjoint complete ordered partitionable (COP) transversals and sets of disjoint holey ordered partitionable (HOP) transversals for Room frames. Our main results include upper and lower bounds on the number of disjoint COP transversals and the number of disjoint HOP transversals for Room frames of type 2n. This work is motivated by the large number of applications of these designs

On the minisymposium problem

The generalized Oberwolfach problem asks for a factorization of the complete graph $K_v$ into prescribed $2$-factors and at most a $1$-factor. When all $2$-factors are pairwise isomorphic and $v$ is odd, we have the classic Oberwolfach problem, which was originally stated as a seating problem: given $v$ attendees at a conference with $t$ circular tables such that the $i$th table seats $a_i$ people and ${\sum_{i=1}^t a_i = v}$, find a seating arrangement over the $\frac{v-1}{2}$ days of the conference, so that every person sits next to each other person exactly once. In this paper we introduce the related {\em minisymposium problem}, which requires a solution to the generalized Oberwolfach problem on $v$ vertices that contains a subsystem on $m$ vertices. That is, the decomposition restricted to the required $m$ vertices is a solution to the generalized Oberwolfach problem on $m$ vertices. In the seating context above, the larger conference contains a minisymposium of $m$ participants, and we also require that pairs of these $m$ participants be seated next to each other for $\left\lfloor\frac{m-1}{2}\right\rfloor$ of the days. When the cycles are as long as possible, i.e.\ $v$, $m$ and $v-m$, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when $v \equiv m \equiv 2 \pmod 4$ and all cycle lengths are even. In addition, we provide extensive results in the case where all cycle lengths are equal to $k$, solving all cases when $m\mid v$, except possibly when $k$ is odd and $v$ is even.Comment: 25 page

The completion of optimal (3,4)(3,4)-packings

A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of quadruples $d(3,4,n)$ denotes the number of blocks in a maximum $3$-$(n,4,1)$ packing design, which is also the maximum number $A(n,4,4)$ of codewords in a code of length $n$, constant weight $4$, and minimum Hamming distance 4. In this paper the undecided 21 packing numbers $A(n,4,4)$ are shown to be equal to Johnson bound $J(n,4,4)$ $( =\lfloor\frac{n}{4}\lfloor\frac{n-1}{3}\lfloor\frac{n-2}{2}\rfloor\rfloor\rfloor)$ where $n=6k+5$, $k\in \{m:\ m$ is odd, $3\leq m\leq 35,\ m\neq 17,21\}\cup \{45,47,75,77,79,159\}$

Understanding the contribution of mode area and slow light to the effective Kerr nonlinearity of waveguides

We resolve the ambiguity in existing definitions of the effective area of a waveguide mode that have been reported in the literature by examining which definition leads to an accurate evaluation of the effective Kerr nonlinearity. We show that the effective nonlinear coefficient of a waveguide mode can be written as the product of a suitable average of the nonlinear coefficients of the waveguide’s constituent materials, the mode’s group velocity and a new suitably defined effective mode area. None of these parameters on their own completely describe the strength of the nonlinear effects of a waveguide.Shahraam Afshar V., T. M. Monro, and C. Martijn de Sterk

Enumeration of Matchings: Problems and Progress

This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley), Mathematical Science Research Institute publication #37, Cambridge University Press, 199

Pairwise balanced designs covered by bounded flats

We prove that for any $K$ and $d$, there exist, for all sufficiently large admissible $v$, a pairwise balanced design PBD$(v,K)$ of dimension $d$ for which all $d$-point-generated flats are bounded by a constant independent of $v$. We also tighten a prior upper bound for $K = \{3,4,5\}$, in which case there are no divisibility restrictions on the number of points. One consequence of this latter result is the construction of latin squares `covered' by small subsquares