18 research outputs found
A Generalization of Plexes of Latin Squares
A -plex of a latin square is a collection of cells representing each row,
column, and symbol precisely times. The classic case of is more
commonly known as a transversal. We introduce the concept of a -weight, an
integral weight function on the cells of a latin square whose row, column, and
symbol sums are all . We then show that several non-existence results about
-plexes can been seen as more general facts about -weights and that the
weight-analogues of several well-known existence conjectures for plexes
actually hold for -weights
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given