Notes on Hamiltonian threshold and chain graphs

TL;DR

This paper revisits and reformulates conditions for Hamiltonicity in threshold graphs, extends these results to chain graphs, and identifies the chain graph with the fewest Hamilton cycles among Hamiltonian chain graphs.

Contribution

It provides new structural characterizations of Hamiltonian threshold and chain graphs and extends classical results to a broader class of chain graphs.

Findings

Reformulated Hamiltonian conditions for threshold graphs.

Extended conditions to chain graphs.

Identified chain graph with minimum Hamilton cycles.

Abstract

We revisit results obtained in [F. Harary, U. Peled, Hamiltonian threshold graphs, Discrete Appl.~Math., 16 (1987), 11--15], where several necessary and necessary and sufficient conditions for a connected threshold graph to be Hamiltonian were obtained. We present these results in new forms, now stated in terms of structural parameters that uniquely define the threshold graph and we extend them to chain graphs. We also identify the chain graph with minimum number of Hamilton cycles within the class of Hamiltonian chain graphs of a given order.