9 research outputs found
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
Starter sequences: generalizations and applications
In this thesis we introduce new types of starter sequences, pseudo-starter sequences,
starter-labellings, and generalized (extended) starter sequences. We apply these new
sequences to graph labeling. All the necessary conditions for the existence of starter,
pseudo-starter, extended, m-fold, excess, and generalized (extended) starter sequences
are determined, and some of these conditions are shown to be sufficient. The relationship between starter sequences and graph labellings is introduced. Moreover, the starter-labeling and the minimum hooked starter-labeling of paths, cycles, and k-
windmills are investigated. We show that all paths, cycles, and k-windmills can be
starter-labelled or minimum starter-labelled
Disjoint skolem-type sequences and applications
Let D = {i₁, i₂,..., in} be a set of n positive integers. A Skolem-type sequence
of order n is a sequence of i such that every i ∈ D appears exactly twice in the
sequence at position aᵢ and bᵢ, and |bᵢ - aᵢ| = i. These sequences might contain
empty positions, which are filled with 0 elements and called hooks. For example,
(2; 4; 2; 0; 3; 4; 0; 3) is a Skolem-type sequence of order n = 3, D = f2; 3; 4g and two
hooks. If D = f1; 2; 3; 4g we have (1; 1; 4; 2; 3; 2; 4; 3), which is a Skolem-type sequence
of order 4 and zero hooks, or a Skolem sequence.
In this thesis we introduce additional disjoint Skolem-type sequences of order n
such as disjoint (hooked) near-Skolem sequences and (hooked) Langford sequences.
We present several tables of constructions that are disjoint with known constructions
and prove that our constructions yield Skolem-type sequences. We also discuss the
necessity and sufficiency for the existence of Skolem-type sequences of order n where
n is positive integers
Existence and embeddings of partial Steiner triple systems of order ten with cubic leaves
Denote the set of 21 non-isomorphic cubic graphs of order 10 by L. We first determine precisely which L is an element of L occur as the leave of a partial Steiner triple system, thus settling the existence problem for partial Steiner triple systems of order 10 with cubic leaves. Then we settle the embedding problem for partial Steiner triple systems with leaves L is an element of L. This second result is obtained as a corollary of a more general result which gives, for each integer v greater than or equal to 10 and each L is an element of L, necessary and sufficient conditions for the existence of a partial Steiner triple system of order v with leave consisting of the complement of L and v - 10 isolated vertices. (C) 2004 Elsevier B.V. All rights reserved