  We present progress on two old conjectures about longest cycles in graphs.
The first conjecture, due to Thomassen from 1978, states that apart from a
finite number of exceptions, all connected vertex-transitive graphs contain a
Hamiltonian cycle. The second 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 both of the aforementioned conjectures: First, we show
that every connected vertex-transitive graph on $n\geq 3$ vertices contains a
cycle of length at least $\Omega(n^{13/21})$, improving on $\Omega(n^{3/5})$
from [De Vos, 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, J. Combin. Theory, Ser.~ B, 1998]. Our proof combines combinatorial
arguments, computer-search and linear programming.
