Chvátal's Condition cannot hold for both a graph and its complement

Chvátal’s Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d1, . . . , dn in nondecreasing order, i < n/2 implies that di > i or dn−i ≥ n − i. We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2. This note is motivated by a discussion in the book of Palmer [7, page 81–85]. A theorem is strong if the conclusion is satisfied only when the hypothesis is satisfied, because then the hypotheses cannot be weakened. Palmer defines the strength of a theorem to be the probability that its hypotheses hold divided by the probability that its conclusion holds. We use the standard random graph model for generating n-vertex simple graphs: the vertex set is {1, . . . , n}, and edge ij occurs with probability p, independently of other edges. Let Gn,p denote the random variable for the resulting graph. In general, “Gn,p ∗Supported in part by the NSF under Award No. DMS-0099608. †Supported in part by the NSA under Award No. MDA904-03-1-0037. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation.