A note on the Song–Zhang theorem for Hamiltonian graphs
Volume 120 / 2010
Colloquium Mathematicum 120 (2010), 63-75
MSC: 05C38, 05C45.
DOI: 10.4064/cm120-1-5
Abstract
An independent set $S$ of a graph $G$ is said to be essential if $S$ has a pair of vertices that are distance two apart in $G$. In 1994, Song and Zhang proved that if for each independent set $S$ of cardinality $k+1$, one of the following condition holds:
(i) there exist $u \neq v \in S$ such that $d(u) + d(v) \geq n$ or $|N(u) \cap N(v)| \geq \alpha (G)$;
(ii) for any distinct $u$ and $v$ in $S$, $|N(u) \cup N(v)| \geq n - \max \{d(x): x \in S\}$,
then $G$ is Hamiltonian. We prove that if for each essential independent set $S$ of cardinality $k+1$, one of conditions (i) or (ii) holds, then $G$ is Hamiltonian. A number of known results on Hamiltonian graphs are corollaries of this result.
