arXiv:2610.35976v1 Announce Type: new Abstract: Let $p>q>5$ be primes. We determine the socles and corresponding actions of quasiprimitive imprimitive groups of degree $2pq$, all of which are almost simple. Using this classification, we prove that a connected graph of order $2pq$ admitting a quasiprimitive automorphism group contains a Hamilton cycle, except possibly for three specified families of actions with socle $\PSL(2,p)$ or $\PSL(2,q^2)$.
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