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 \(e_2\).