A matching M in a graph G is connected if the induced subgraph on the vertex set \(V(\{e,f\})\) is connected for every pair of edges e, f in M. The problem of finding 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. Füredi 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 Füredi et al.’s conjecture, and then using these properties, we prove that Füredi et al.’s conjecture holds for \(t\le 22\) .
