Connected Turán number of trees

The connected Turán number is a variant of the much studied Turán number, ex(n,F), the largest number of edges that an n-vertex F-free graph may contain. We start a systematic study of the connected Turán number exc(n,F), the largest number of edges that an n-vertex connected F-free graph may contain. We focus on the case where the forbidden graph is a tree. Prior to our work, exc(n,T) was determined only for the case T is a star or a path. Our main contribution is the determination of the exact value of exc(n,T) for small trees, in particular for all trees with at most six vertices, as well as some trees on seven vertices and several infinite families of trees. We also collect several lower-bound constructions of connected T-free graphs based on different graph parameters. The celebrated conjecture of Erdős and Sós states that for any tree T, we have ex(n,T) ≤ (|T|−2)n/2. We address the problem how much smaller exc(n,T) can be, what is the smallest possible ratio of exc(n,T) and (|T|−2)n/2 as |T| grows.