Lovász's Hamilton-path conjecture for connected vertex-transitive graphs

Let $G$ be a connected vertex-transitive graph, meaning that for every $u,v in V(G)$ there is an automorphism of $G$ mapping $u$ to $v$. A Hamilton path is a path containing every vertex of $G$.

Lovász's conjecture. Every connected vertex-transitive graph has a Hamilton path.

This classical conjecture asks whether vertex-transitivity guarantees a spanning path; it remains open in full generality, although substantial results are known for dense graphs and long cycles in general vertex-transitive graphs.