A step forward is made in a long standing Lovász problem regarding existence of Hamilton paths in vertex-transitive graphs. It is shown that a vertex-transitive graph of order a product of two primes arising from a primitive action of PSL(2;p) on the cosets of a subgroup isomorphic to D_p−1 has a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chvátal's theorem and Jackson's theorem, to certain results from matrix algebra, graph quotienting, and polynomial representations of quadratic residues in terms of primitive roots in finite fields. Also, Hamilton cycles are proved to exist in vertex-transitive graphs of order a product of two primes arising from a primitive action of either PΩ(2d; 2), M₂₂, A₇, PSL(2; 13), or PSL(2; 61). The results of this paper, combined together with other known results, imply that all connected vertex-transitive graphs of order a product of two primes, except for the Petersen graph, have a Hamilton cycle.

在关于顶点传递图中哈密顿路径的存在的长期存在的 Lovász 问题中向前迈进了一步。结果表明，由 PSL(2; p ) 在同构于D _{p -1}的子群的陪集上产生的两个素数的乘积阶顶点传递图具有哈密顿圈。证明中使用的基本工具范围从关于存在哈密顿圈的经典结果，如 Chvátal 定理和杰克逊定理，到矩阵代数、图商和二次残差的多项式表示的某些结果（根据有限域中的原始根）。此外，哈密顿圈被证明存在于阶的顶点传递图中，该阶是两个素数的乘积，这些素数是由其中一个的原始动作产生的PΩ (2 d；2)、M ₂₂、A ₇、PSL(2；13) 或 PSL(2；61)。本文的结果与其他已知结果相结合，表明除了彼得森图外，所有阶为两个素数乘积的连通顶点传递图都有一个哈密顿圈。