Embedding Partial Steiner Triple Systems with Few Triples

Horsley, Daniel

doi:10.1137/130939365

Cited by 5 publications

(6 citation statements)

References 20 publications

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“…Some results along these lines were obtained in [10,4,7,25,26]. In particular, Horsley [25] showed that if F has at most n 2 /50 − o(n 2 ) blocks then it has an embedding of order 8n/5 + O (1). In this paper we determine an asymptotically optimal bound for embeddings of relatively sparse partial STSs.…”

mentioning

confidence: 85%

“…As r ≤ εn 2 and |V (G * )| = n + t − |B| > n, for ε sufficiently small compared to k we have 2k We now present a construction showing that Lemma 3.3 is tight for r > 2n up to a multiplicative constant, which also translates to sharpness of Theorem 1.1 and Theorem 1.2. To remind the reader, a linear lower bound on r is not a coincidence: for k = 3 and r < n/2, Horsley [25] conjectured that any partial STS of order n has an embedding of the same order, and it is natural to believe that a similar result should hold for partial (n, k)-designs as well. Let k ≥ 3 and 2n < r ≤ 4n 2 /k 2 .…”

Section: Completing Steiner Triple Systems and Other Block Designsmentioning

confidence: 99%

“…While the bound 2n + 1 is sharp in general, it is a natural question whether this can be improved if F is sparse, that is if F contains only a few blocks. Some results along these lines were obtained in [10,4,7,25,26]. In particular, Horsley [25] showed that if F has at most n 2 /50 − o(n 2 ) blocks then it has an embedding of order 8n/5 + O (1).…”

mentioning

confidence: 95%

“…We present a construction which shows this in Section 3. For |F| < n/2 and k = 3, Horsley [25] conjectured that in fact there exists an embedding of F which is of the same order as F.…”

mentioning

confidence: 99%

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Completion and deficiency problems

Nenadov¹,

Sudakov²,

Wagner³

2019

Preprint

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Given a partial Steiner triple system (STS) of order n, 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 n with at most r ≤ εn 2 triples, it can always be embedded into a complete STS of order n + O( √ r), 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 P and a graph G, we define the deficiency def(G) of the graph G with respect to the property P to be the smallest positive integer t such that the join G * Kt has property P. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a K k -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.

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mentioning

confidence: 85%

Section: Completing Steiner Triple Systems and Other Block Designsmentioning

confidence: 99%

mentioning

confidence: 95%

“…We present a construction which shows this in Section 3. For |F| < n/2 and k = 3, Horsley [25] conjectured that in fact there exists an embedding of F which is of the same order as F.…”

mentioning

confidence: 99%

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Completion and deficiency problems

Nenadov¹,

Sudakov²,

Wagner³

2019

Preprint

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“…Proof By [10, Theorem 5.2], if

G'

is an even graph of order

a

such that

a ⩾ 103, false| E false(G' false) false| \equiv 0 (\mod 3), false| E false(G' false) false| ⩾ false(\frac{0.0pt}{a} false) - \frac{1}{128} false(3 a^{2} - 54 a - 409 false)

and at least

\frac{1}{8} (3 a + 17)

vertices of

G'

have degree

a - 1

, then there is a

K_{3}

‐decomposition of

G'

. Let

a = false| A false|

and let

G' = K_{A} - (\overset{K_{Z}}{true} \lor G)

.…”

Section: Hardness Of Finding Small Embeddings Of Specified Ordersmentioning

confidence: 99%

On determining when small embeddings of partial Steiner triple systems exist

Bryant

Gunasekara

Horsley

2020

J of Combinatorial Designs

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A partial Steiner triple system of order u is a pair (U, A) where U is a set of u elements and A is a set of triples of elements of U such that any two elements of U occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system (U, A) is a (complete) Steiner triple system (V, B) such that U ⊆ V and A ⊆ B. For a given partial Steiner triple system of order u it is known that an embedding of order v 2u + 1 exists whenever v satisfies the obvious necessary conditions. Determining whether "small" embeddings of order v < 2u + 1 exist is a more difficult task. Here we extend a result of Colbourn on the NP-completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist.

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