The Lovász conjecture (in its most widely encountered form) states that without exception, every connected vertex-transitive graph is traceable (Lovász 1970; cf. Gould 1991, p. 45; Mütze 2024). Amusingly, Babai (1979, 1996) published a directly contradictory conjecture. A similar conjecture attributed to Thomassen asserts that, with exactly five exceptions (the nonhamiltonian vertex-transitive graph), every vertex-transitive graph is Hamiltonian. While the Lovász conjecture has subsequently been verified for several special orders and classes, both conjectures remain open.

Hamiltonian cycle | Hamiltonian graph | Hamiltonian path | nonhamiltonian vertex-transitive graph | traceable graph | vertex-transitive graph