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.

References

Conjecture 1.6. Every connected vertex-transitive graph contains a Hamilton path, and, except for five known examples, a Hamilton cycle.

— Hamiltonicity of expanders: optimal bounds and applications  (2402.06603 - Draganić et al., 2024) in Conjecture 1.6, Section 1.1

Another central open problem concerning Hamiltonicity is the Lov\'asz conjecture for connected vertex-transitive graphs. The strongest natural statement, that every connected vertex-transitive graph has a Hamilton cycle, is false. Lov\'asz asked whether there exists a connected vertex-transitive graph without a Hamilton path, while a stronger conjecture, commonly attributed to Thomassen, asserts that all but finitely many connected vertex-transitive graphs have a Hamilton cycle; see.

— Thinning and sprinkling: from robust sampling to almost Hamiltonicity  (2609.30165 - Christoph et al., 24 Sep 2026) in Section 1, subsection “Almost Hamilton cycles”

Another central open problem concerning Hamiltonicity is the Lov\'asz conjecture for connected vertex-transitive graphs. The strongest natural statement, that every connected vertex-transitive graph has a Hamilton cycle, is false. Lov\'asz asked whether there exists a connected vertex-transitive graph without a Hamilton path, while a stronger conjecture, commonly attributed to Thomassen, asserts that all but finitely many connected vertex-transitive graphs have a Hamilton cycle; see.

— Thinning and sprinkling: from robust sampling to almost Hamiltonicity  (2609.30165 - Christoph et al., 24 Sep 2026) in Section 1, subsection “Almost Hamilton cycles”

Both conjectures remain wide open in general, but sufficiently high degree is known to guarantee Hamiltonicity.

— Longest cycles intersect linearly in highly connected graphs  (2609.20724 - Ma et al., 17 Sep 2026) in Section 6.2, “Implications for vertex-transitive graphs”

Both conjectures remain wide open in general, but sufficiently high degree is known to guarantee Hamiltonicity.

— Longest cycles intersect linearly in highly connected graphs  (2609.20724 - Ma et al., 17 Sep 2026) in Section 6.2, “Implications for vertex-transitive graphs”

It remains open whether every connected vertex-transitive graph on $n$ vertices admits a cycle of length $(1-o(1))n$.

— Linear circumference in vertex-transitive graphs  (2610.02053 - Ma et al., 1 Oct 2026) in Section 5, “Concluding remarks”