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

Abstract

Chvatal’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 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 almost always satisfies Q” means that the probability of Gn,p satisfying Q tends to 1 as n→∞. We restrict our attention to constant 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.