On Three Conjectures of GRAFFITI

In this paper, we consider three conjectures of the computer program GRAFFITI. Moreover, we prove that every connected graph with minimum degree \(\delta\) and diameter \(d_m\) contains a matching of size at least \(\frac{\delta(d_m + 1)}{6}\). This inequality improves one of the conjectures under the additional assumption that \(\delta \geq 6\).