An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision

Abstract

Smith's conjecture asserts that in every $k$-connected graph with $k\geq 2$, any two longest cycles intersect in at least $k$ vertices. In this work, we establish an $\Omega(k^{8/11})$ bound for this conjecture, improving upon the $\Omega(k^{2/3})$ bound of Ma and Zhao. Our proof combines a Ramsey theoretic refinement of the traditional Tur\'an-type approach with computer search.