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A step forward is made in a long standing Lov\'{a}sz's problem regardinghamiltonicity of vertex-transitive graphs by showing that every connectedvertex-transitive graph of order a product of two primes, other than thePetersen graph, contains a Hamilton cycle. Essential tools used in the proofrange from classical results on existence of Hamilton cycles, such asChv\'atal's theorem and Jackson's theorem, to certain results on polynomialrepresentations of quadratic residues at primitive roots in finite fields.

Hamilton cycles in vertex-transitive graphs of order a product of two primes