Hamilton cycles with rotational symmetry in connected vertex-transitive graphs

Motivated by recently introduced graph parameter that quantifies how symmetric a Hamilton cycle in a graph can be the proposed project considers symmetries of Hamilton cycles in connected vertex-transitive graphs. The parameter was introduced in 2022 by Gregor, Merino and Muetze. Formally, let X be a graph with n vertices. We say that a Hamilton cycle C=(v0,...,vn-1) is k-symmetric if the mapping f: V(X) -> V(X) defined by f(vi)=vi+n/k for all i=0,...,n-1, where indices are considered modulo n, is an automorphism of X. In this case we have C=(P,f (P), f 2(P),...,f k-1(P)) for the path P=v0,...,vn/k-1.  Therefore the entire cycle C can be reconstructed from the path P, which contains only a 1/k-fraction of all the vertices, by repeatedly applying the automorphism f to it. If we lay out the vertices equidistantly on a circle, and draw edges of X as straight lines, then we obtain a drawing of X with k-fold rotational symmetry, that is, f is a rotation by 360/k degrees. The maximum k for which the Hamilton cycle C of X is k-symmetric is called the compression factor of C and is denoted by kappa(X,C). For a graph X we define kappa(X)=max{kappa(X,C) | C is a Hamilton cycle in X}, and we refer to this quantity as the Hamilton compression of X. If X has no Hamilton cycle, then we define k(X)=0. The quantity kappa(X) can be therefore seen as a measure for the nicest (that is, the most symmetric) way of drawing the graph X on a circle. The main object of the proposed project is to study Hamilton compression of connected vertex-transitive graphs, those for which the existence of Hamilton cycles is already known, and in view of the connection with the polycirculant conjecture, also those for which no information on Hamilton cycles has been obtained thus far. In this sense the proposed project has a clear link to the Lovasz hamiltonicity problem for vertex-transitive graphs.