On deficiency problems for graphs

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