Hamiltonicity of certain vertex-transitive graphs revisited

Abstract

Motivated by Gregor et al. (2023) [7], existence of Hamilton cycles, admitting large rotational symmetry, in certain vertex-transitive graphs is investigated. Given a graph X with a Hamilton cycle C, the compression factor $\kappa (X,C)$ of C is the order of the largest cyclic subgroup of $\text{Aut}\left(C\right)\cap \text{Aut}\left(X\right)$, and the Hamilton compression $\kappa \left(X\right)$ of X is the maximum compression factor over all of its Hamilton cycles. It is shown that for $p,q$ distinct primes, vertex-primitive graphs of order pq have Hamilton compression equal to p or q. In addition, for each $n=1+2^{2^e}$, $e>1$, a connected vertex-transitive graph of order 3n and Hamilton compression equal to n is constructed. As a consequence Hamilton compressions of vertex-transitive graphs of order 3p, p a prime, are determined. Similarly, Hamilton compressions of vertex-transitive graphs of order 2p, p a prime, are also computed.