Lovász’s conjecture on Hamiltonicity of vertex-transitive graphs

Establish that every connected vertex-transitive graph contains a Hamilton path and, except for five known examples, a Hamilton cycle.

Background

Vertex-transitive graphs generalize symmetry-rich structures including Cayley graphs. Lovász’s conjecture (1969) is among the central open problems in Hamiltonicity, implying strong structural connectivity and spanning cycle existence in this class.

The paper mentions this conjecture as motivation for studying Hamiltonicity in pseudorandom and Cayley graphs; it remains open in full generality.