2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75629This paper presents a solution of the polycirculant conjecture which states that every vertex-transitive graph G has an automorphism that permutes the vertices in cycles of the same length. This is done by identifying vertex-transitive graphs as coset graphs. For a coset graph H, an equivalence relation $\sim$ is defined on the vertices of cosets with classes as double cosets of the stabiliser and any other proper subgroup A' of a transitive group A of G. Induced left translations of elements of the subgroup A' are semi-regular since they preserve these double cosets and acts regularly on each of them. The coset graph is equivalent to G by a theorem of Sabidussi.6 pagesCombinatoricsGroup Theory05C25;20B25;05E15;20F65text