On Hamilton cycles in connected vertex-transitive graphs of order $2pq$

The existence of Hamilton cycles in connected vertex-transitive graphs is a core open problem in algebraic graph theory, originating from Lov'asz’s 1969 conjecture. All connected vertex-transitive graphs of order $pq$ are known to be Hamiltonian except the Petersen graph, and primitive graph of order $2pq$ are resolved except the Coxeter graph. This paper considers connected vertex-transitive graphs of order $2pq$ where every transitive automorphism subgroup admits a maximal intransitive normal subgroup inducing prime-length orbits. We prove that all such graphs contain a Hamilton cycle, with no new exceptions beyond the already characterized non-qualifying graphs. This result covers a large non-quasiprimitive graphs of order $2pq$, advancing the full resolution of the $2pq$.