Almost all graphs are rigid - Revisited

A graph is called asymmetric if it has the identity mapping as its only automorphism. In [P. Erdõs, A. Rényi, Asymmetric Graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315], P. Erdõs and A. Rényi have proven that almost all graphs are asymmetric. A graph is called rigid if it has the identity mapping as its only endomorphism, which is a stronger property than asymmetry. By adopting the approach of Erdõs and Rényi, it is shown that almost all graphs are rigid. A different proof of that result has already been published in [V. Koubek, V. Rödl, On the Minimum Order of Graphs with Given Semigroup, J. Combin. Theory Ser. B 36 (1984) 135–155] (as well as in [P. Hell, J. Nešetřil, Graphs and Homomorphisms, Oxford U. Press, Oxford, 2004]).