On deficiency problems for graphs

TL;DR

This paper fully resolves the problem of determining the minimum edges needed in an n-vertex graph to ensure its join with a complete graph contains a perfect K_r-factor, extending to bipartite graphs of bounded degree and small bandwidth.

Contribution

The paper provides a complete solution to the deficiency problem for graphs regarding K_r-factors and extends results to bipartite graphs with bounded degree and small bandwidth.

Findings

Determined the minimum edges needed for G*K_t to contain a K_r-factor.

Extended results to bipartite graphs with bounded degree and small bandwidth.

Resolved the deficiency problem fully for the specified properties.

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 non-negative 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.