The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture

From Theorem 1.1 of the Hadwiger paper: deleting at most three vertices from its graphs gives graphs with $\alpha=2$ on $4t-1$ vertices whose connected-matching number stays below $t$ for all large $t$, so the sharp conjecture fails. The weaker conjecture of the same authors, that $\mathrm{cm}(G)\ge c_0|V(G)|$ for some absolute $c_0>0$, is not settled: the bound $\mathrm{cm}<m/100$ leaves smaller coefficients open, and the paper says so.

From Theorem 1.1 of the Hadwiger paper: deleting at most three vertices from its graphs gives graphs with $\alpha=2$ on $4t-1$ vertices whose connected-matching number stays below $t$ for all large $t$, so the sharp conjecture fails. The weaker conjecture of the same authors, that $\mathrm{cm}(G)\ge c_0|V(G)|$ for some absolute $c_0>0$, is not settled: the bound $\mathrm{cm}<m/100$ leaves smaller coefficients open, and the paper says so.