Intersections of longest cycles in vertex-transitive and highly connected graphs

Abstract: Motivated by the classical conjectures of Lov\'asz, Thomassen, and Smith, recent work has renewed interest in the study of longest cycles in important graph families, such as vertex-transitive and highly connected graphs. In particular, Groenland et al.\ proved that if two longest cycles and in a graph share $m$ vertices, then there exists a vertex cut of size $O(m^{8/5})$ separating them, yielding improved bounds toward these conjectures. Their proof combines Tur\'an-type arguments with computer-assisted search. We prove two results addressing problems of Babai (1979) and Smith (1984) on intersections of longest cycles in vertex-transitive and highly connected graphs. First, we strengthen the bound of Groenland et al.\ by showing that if two longest cycles and in a graph share $m$ vertices, then there exists a vertex cut of size $O(m^{3/2})$ separating them. As a consequence, we show that in every (k)-connected graph, any two longest cycles intersect in at least (\Omega(k^{2/3})) vertices, improving the best known bound toward Smith's conjecture. Our proof is purely combinatorial, employing supersaturation-type estimates beyond the existing Tur\'an-type approach. Second, we prove that in every connected vertex-transitive graph on (n) vertices, any two longest cycles intersect in at least (f(n)) vertices for some function (f(n)\to\infty) as (n\to\infty), thereby resolving a problem of Babai (1979) for the class of vertex-transitive graphs central to his original motivation. In doing so, we introduce a new method for constructing longer cycles in vertex-transitive graphs based on a given cycle, which may be of independent interest.