Longest cycles in vertex-transitive and highly connected graphs

Abstract

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle (and hence path) of length at least $\Omega(n^{13/21})$, improving on $\Omega(n^{3/5})$ from [DeVos, \emph{arXiv:2302:04255}, 2023]. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $\Omega(r^{5/8})$ vertices, improving on $\Omega(r^{3/5})$ from [Chen, Faudree and Gould, \emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.