Completion and deficiency problems

Nenadov, Rajko; Sudakov, Benny; Wagner, Adam Zsolt

doi:10.1016/j.jctb.2020.05.004

Cited by 12 publications

(28 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A natural question dating back to the 1970s asks for the order of the smallest complete Steiner triple system a fixed partial Steiner triple system can be embedded into (see e.g. [3,10,11]). Similarly, there has been interest in establishing the order of the smallest Latin square that a fixed partial Latin square can be embedded into (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, there has been interest in establishing the order of the smallest Latin square that a fixed partial Latin square can be embedded into (see e.g. [5,6,11]).…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by these research directions, Nenadov, Sudakov and Wagner [11] introduced the notion of graph deficiency: for a graph G and integer t ≥ 0, denote by G * K t the join of G and K t , which is the graph obtained from G by adding t new vertices and adding all edges incident to at least one of the new vertices. Given a global spanning property P and a graph G, the deficiency def(G) of the graph G with respect to the property P is the smallest t ≥ 0 such that the join G * K t has property P.…”

Section: Introductionmentioning

confidence: 99%

“…One of the main results in [11] is a bound on def(G) with respect to the Hamiltonicity property for graphs G of a given density. More precisely, the following result answers the question of how many edges an n-vertex graph G can have such that G * K t does not contain a Hamilton cycle.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On deficiency problems for graphs

Freschi¹,

Hyde²,

Treglown³

2021

Preprint

View full text Add to dashboard Cite

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property P and a graph G, the deficiency def(G) of the graph G with respect to the property P is the smallest non-negative integer t such that the join G * Kt has property P. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph G needs to ensure G * Kt contains a Kr-factor (for any fixed r ≥ 3). In this paper we resolve their problem fully. We also give an analogous result which forces G * Kt to contain any fixed bipartite (n + t)-vertex graph of bounded degree and small bandwidth.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Similarly, there has been interest in establishing the order of the smallest Latin square that a fixed partial Latin square can be embedded into (see e.g. [5,6,11]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On deficiency problems for graphs

Freschi¹,

Hyde²,

Treglown³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…This paper concerns the problem of determining whether a given partial Steiner triple system has a small embedding of a specified order. Various aspects of this problem have been addressed in many papers (see [2,3,4,10,14], eg). In this paper, we provide updates on two of these contributions, namely [4] and [2].…”

Section: Abstract Embeddings Np-completeness Partial Steiner Triple Systems 1 | Introductionmentioning

confidence: 99%

On determining when small embeddings of partial Steiner triple systems exist

Bryant

Gunasekara

Horsley

2020

J of Combinatorial Designs

View full text Add to dashboard Cite

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.

show abstract

Covering 3‐uniform hypergraphs by vertex‐disjoint tight paths

Han

2022

Journal of Graph Theory

View full text Add to dashboard Cite

For α > 0 and large integer n, let H be an n-vertex 3uniform hypergraph such that every pair of vertices isedges. We show that H contains two vertex-disjoint tight paths whose union covers the vertex set of H . The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the deficiency problems, recently introduced by Nenadov, Sudakov, and Wagner: every such H can be made Hamiltonian by adding at most two vertices and all triples intersecting them.

show abstract

Completion and deficiency problems

Cited by 12 publications

References 35 publications

On deficiency problems for graphs

On deficiency problems for graphs

On determining when small embeddings of partial Steiner triple systems exist

Covering 3‐uniform hypergraphs by vertex‐disjoint tight paths

Contact Info

Product

Resources

About