On deficiency problems for graphs
Abstract
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 $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest nonnegative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$factor (for any fixed $r\geq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$vertex graph of bounded degree and small bandwidth.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.04389
 arXiv:
 arXiv:2102.04389
 Bibcode:
 2021arXiv210204389F
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 12 pages, author accepted manuscript, to appear in Combinatorics, Probability and Computing