Intersections of largest bonds in [formula omitted]-connected graphs

Abstract

S. Smith conjectured that any two distinct longest cycles of a k -connected graph must meet in at least k vertices when k ≥ 2 . Here we provide evidence for the dual version of the Smith conjecture: if C and D are largest bonds in a k -connected graph G , then the number of components in G − ( C ∪ D ) is at least k + 2 − | C ∩ D | . Both conjectures have been proven only for small k . This article establishes a linear lower bound on the number of components of G − ( C ∪ D ) which holds for all values of k ≥ 7 . This is stronger than the best bound established to date for Smith’s original conjecture.