Hamilton Cycles In Vertex-Transitive Graphs of Order 10p

Huye Chen, Jingjian Li, Hao Yu

Published 2025-06-24Version 1

After a long term efforts, the Hamiltonian problem of connected vertex-transitive graphs of order $pq$ (where $p$ and $q$ are primes) was finally finshed in 2021, see [10]. Fifteen years ago, mathematicians began to challenge this problem for graphs of order $2pq$. Among of them, it was proved in 2012 (see [21]) that every connected vertex-transitive graph of order $10p$ (where $p\neq7$ is a prime) contains a Hamilton path, with the exception of a family of graphs which was recently confirmed in [11]. In this paper, a further conclusion will be achieved: every connected vertex-transitive graph of order $10p$ (where $p$ is a prime) contains a Hamilton cycle, except for the truncation of the Petersen graph.