On deficiency problems for graphs

Andrea Freschi, Joseph Hyde, Andrew Treglown

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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 that forces G∗Kt to contain any fixed bipartite (n+t) -vertex graph of bounded degree and small bandwidth.
Original languageEnglish
Pages (from-to)478-488
Number of pages11
JournalCombinatorics, Probability and Computing
Issue number3
Early online date27 Sept 2021
Publication statusPublished - May 2022

Bibliographical note

Funding Information:
The authors are grateful to the referee for a helpful and careful review. Joseph Hyde was supported by the UK Research and Innovation Future Leaders Fellowship MR/S016325/1.

Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.


  • graph deficiency
  • clique factors
  • bandwidth theorems

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Applied Mathematics
  • Statistics and Probability
  • Computational Theory and Mathematics


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