Connected matching in graphs with independence number two

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $\alpha(G)=2$ is closely related to Hadwiger's conjecture for graphs with independence number 2. The problem of finding a large connected matching in a general graph is NP-hard. F{\"u}redi et al. in 2005 conjectured that each $(4t-1)$-vertex graph $G$ with $\alpha(G)=2$ contains a connected matching of size at least $t$. Cambie recently showed that if this conjecture is false, then so is Hadwiger's conjecture. In this paper, we present a number of properties possessed by a counterexample to F{\"u}redi et al.'s conjecture, and then using these properties, we prove that F{\"u}redi et al.'s conjecture holds for $t\leq22$.