Toughness of Graphs and $2$-Factors with Given Properties

Abstract

Let $G$ be a $2$-tough graph on at least five vertices and let $e_1,e_2$ be a pair of arbitrarily given edges of $G$. Then
(a) There exists a $2$-factor in G containing $e_1,e_2$.
(b) There exists a $2$-factor in G avoiding $e_1,e_2$.
(c) There exists a $2$-factor in G containing $e_1$ and avoiding $e_2$.