A generalization of a theorem of Nash-Williams

Abstract

Chvátal (J Combin Theory Ser B 12:163–168, 1972) gave a well-known sufficient condition for a graphical sequence to be forcibly hamiltonian, and showed that in some sense his condition is best possible. Nash-Williams (Recent Trends in Graph Theory. Springer, Berlin, pp. 197–210, 1971) gave examples of forcibly hamiltonian n-sequences that do not satisfy Chvátal’s condition, for every \(n\ge 5\). In this note we generalize the Nash-Williams examples, and use this generalization to generate \(\varOmega \left( \frac{2^n}{\sqrt{n}} \right) \) forcibly hamiltonian n-sequences that do not satisfy Chvátal’s condition.