A generalization of a theorem of Nash-Williams

TL;DR

This paper generalizes Nash-Williams' examples to produce a large class of forcibly hamiltonian sequences that do not meet Chvatal's condition, highlighting limitations of the condition's sufficiency.

Contribution

It extends Nash-Williams' construction to generate exponentially many forcibly hamiltonian sequences violating Chvatal's condition.

Findings

Generated (2^n/n^{0.5}) sequences

Demonstrated limitations of Chvatal's condition

Provided a broader class of counterexamples

Abstract

In 1972, Chvatal 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 gave examples of forcibly hamiltonian n-sequences that do not satisfy Chvatla's condition for every n at least 5. In this note we generalize the Nash-Williams examples, and use this generalization to generate \Omega(2^n/n^.5) forcibly hamiltonian n-sequences that do not satisfy Chvatal's condition