Cycles Containing a Subset of a Given Set of Elements in Cubic Graphs
Received:June 12, 2012  Revised:September 12, 2012
Key Word: contraction   cubic graphs   cyclability.  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10971027).
Author NameAffiliation
Sheng BAU School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Institute of Discrete Mathematics, Inner Mongolia University of Nationalities, Inner Mongolia 028005, P. R. China 
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      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.
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