Cycles Containing a Subset of a Given Set of Elements in Cubic Graphs

Abstract:

      The technique of contractions and the known results in the study of cycles in $3$-connected cubic graphs are applied to obtain the following result. Let $G$ be a $3$-connected cubic graph, $X\subseteq V(G)$ with $|X| = 16$ and $e\in E(G)$. Then either for every $8$-subset $A$ of $X$, $A\cup\{e\}$ is cyclable or for some $14$-subset $A$ of $X$, $A\cup\{e\}$ is cyclable.