Properties of Hamilton cycles of circuit graphs of matroids

Let G be a circuit graph of a connected matroid. P. Li and G. Liu Comput. Math. Appl., 2008, 55: 654–659] proved that G has a Hamilton cycle including e and another Hamilton cycle excluding e for any edge e of G if G has at least four vertices. This paper proves that G has a Hamilton cycle including e and excluding e′ for any two edges e and e′ of G if G has at least five vertices. This result is best possible in some sense. An open problem is proposed in the end of this paper.