Tight embeddings of partial quadrilateral packings

Let f ( n ; C 4 ) be the smallest integer such that, given any set of edge disjoint quadrilaterals on n vertices, one can extend it into a complete quadrilateral decomposition by including at most f ( n ; C 4 ) additional vertices. It is known, and it is easy to show, that n − 1 ⩽ f ( n ; C 4 ) . Here we settle the longstanding problem that f ( n ; C 4 ) = n + o ( n ) .