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One of the most important open problems in the field of graph colouring oreven graph theory is the conjecture of Hadwiger. This conjecture was theinspiration for many mathematical works, one of them being the work ofF\"uredi, Gy\'arf\'as and Simonyi in which they "risked" to conjecture theprecise bound for a graph with independence number $2$ to contain a certainconnected matching. We prove that their conjecture would be a corollary ofHadwiger's conjecture or equivalently if their risky conjecture would be false,then Hadwiger's conjecture would be false as well.

Hadwiger's conjecture implies a conjecture of F\"uredi-Gy\'arf\'as-Simonyi