Some remarks on universal graphs

A corrected proof is given for the existence of a universal countable {C3, C5, …, C2s+1}-free graph. We also prove that there is a universal countable free graph. There is no universal countable H-free graph if H is the dispoint union of 3 or more complete n-cliques for some n ⩾ 2, plus one vertex, joined to every other point.