  For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ has as
vertices all $k$-element subsets of an $n$-element ground set, and an edge
between any two disjoint sets. It has been conjectured since the 1970s that all
Kneser graphs admit a Hamilton cycle, with one notable exception, namely the
Petersen graph $K(5,2)$. This problem received considerable attention in the
literature, including a recent solution for the sparsest case $n=2k+1$. The
main contribution of this paper is to prove the conjecture in full generality.
We also extend this Hamiltonicity result to all connected generalized Johnson
graphs (except the Petersen graph). The generalized Johnson graph $J(n,k,s)$
has as vertices all $k$-element subsets of an $n$-element ground set, and an
edge between any two sets whose intersection has size exactly $s$. Clearly, we
have $K(n,k)=J(n,k,0)$, i.e., generalized Johnson graph include Kneser graphs
as a special case. Our results imply that all known families of
vertex-transitive graphs defined by intersecting set systems have a Hamilton
cycle, which settles an interesting special case of Lov\'asz' conjecture on
Hamilton cycles in vertex-transitive graphs from 1970. Our main technical
innovation is to study cycles in Kneser graphs by a kinetic system of multiple
gliders that move at different speeds and that interact over time, reminiscent
of the gliders in Conway's Game of Life, and to analyze this system
combinatorially and via linear algebra.
