  An {\it overlap representation} of a graph $G$ assigns sets to vertices so
that vertices are adjacent if and only if their assigned sets intersect with
neither containing the other. The {\it overlap number} $\ol(G)$ (introduced by
Rosgen) is the minimum size of the union of the sets in such a representation.
We prove the following: (1) An optimal overlap representation of a tree can be
produced in linear time, and its size is the number of vertices in the largest
subtree in which the neighbor of any leaf has degree 2. (2) If $\delta(G)\ge 2$
and $G\ne K_3$, then $\ol(G)\le |E(G)|-1$, with equality when $G$ is connected
and triangle-free and has no star-cutset. (3) If $G$ is an $n$-vertex plane
graph with $n\ge5$, then $\ol(G)\le 2n-5$, with equality when every face has
length 4 and there is no star-cutset. (4) If $G$ is an $n$-vertex graph with
$n\ge 14$, then $\ol(G)\le \floor{n^2/4-n/2-1}$, and this is sharp (for even
$n$, equality holds when $G$ arises from $K_{n/2,n/2}$ by deleting a perfect
matching).

Combinatorics