Journal of Combinatorics
Volume 7 (2016)
Guest editors: Rong Luo and Cun-Quan Zhang
Cyclable sets of vertices in 3-connected graphs
Pages: 495 – 508
Given a graph $G$, a set $S \subseteq V (G)$ is called cyclable (in $G$) if $G$ has a cycle containing every vertex of $S$; $G$ is hamiltonian if $V (G)$ is cyclable in $G$. Beginning with a result of Dirac in 1952, many results on sufficient conditions that relate degree sum and neighborhood conditions for hamiltonicity and cyclability, have been obtained. We give a new sufficient condition on degree sums and neighborhoods of any four independent vertices in a graph. We also study the extremal cases of this condition.