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

Abstract

A step forward is made in a long standing Lov\'{a}sz's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph, contains a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chv\'atal's theorem and Jackson's theorem, to certain results on polynomial representations of quadratic residues at primitive roots in finite fields.