On deficiency problems for graphs

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∗K_t 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∗K_t contains a K_r -factor (for any fixed r≥3 ). In this paper, we resolve their problem fully. We also give an analogous result that forces G∗K_t to contain any fixed bipartite (n+t) -vertex graph of bounded degree and small bandwidth.