Cohesion and Non-Separating Trees in Connected Graphs

If \(T\) is a tree on \(n\) vertices, \(n \geq 3\), and if \(G\) is a connected graph such that \(d(u) + d(v) + d(u,v) \geq 2n\) for every pair of distinct vertices of \(G\), it has been conjectured that \(G\) must have a non-separating copy of \(T\). In this note, we prove this result for the special case in which \(d(u) + d(v) + d(u,v) \geq 2n + 2\) for every pair of distinct vertices of \(G\), and improve this slightly for trees of diameter at least four and for some trees of diameter three.