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 that forces
$G*K_t$
to contain any fixed bipartite
$(n+t)$
-vertex graph of bounded degree and small bandwidth.
ON DEFINING SETS IN LATIN SQUARES AND TWO INTERSECTION PROBLEMS, ONE FOR LATIN SQUARES AND ONE FOR STEINER TRIPLE SYSTEMS