Abstract
A crucial step in the Erdos-Renyi (1960) proof that the double-jump threshold is also the planarity threshold for random graphs is shown to be invalid. We prove that whenp=1/n, almost all graphs do not contain a cycle with a diagonal edge, contradicting Theorem 8a of Erdos and Renyi (1960). As a consequence, it is proved that the chromatic number is 3 for almost all graphs whenp=1/n.
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