On two conjectures about the intersection of longest paths and cycles

A conjecture attributed to Smith states that every two longest cycles in ak-connected graph intersect in at leastkvertices. In this paper, we show that every two longest cycles in ak-connected graph onnvertices intersect in at leastmin{n, 8k - n - 16} vertices, which confirms Smith's conjecture when k >=(n + 16)/7. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either k <= 7or k >=(n + 9)/7. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.