Abstract
We prove that connected vertex-transitive digraphs of order $p^{5}$ (where $p$ is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G'$ is generated by two elements or elementary abelian is Hamiltonian.
Highlights
We prove that connected vertex-transitive digraphs of order p5 are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup G of prime power order such that G is generated by two elements or elementary abelian is Hamiltonian
One of the most famous problems in vertex-transitive graphs theory is the problem of existence of Hamilton paths/cycles in finite connected vertex-transitive graphs
The fact that none of these four graphs is a Cayley graph has led to a folklore conjecture that every connected Cayley graph with order greater than 2 has a Hamilton cycle
Summary
One of the most famous problems in vertex-transitive graphs theory is the problem of existence of Hamilton paths/cycles (that is, simple paths/cycles going through all vertices) in finite connected vertex-transitive graphs (or digraphs). As for Cayley graphs, perhaps the biggest achievement on this subject is due to Witte ( Morris) who proved that a connected Cayley digraph of any p-group has a Hamilton cycle [21]. It seems to be quite a challenge to generalize Witte’s theorem on Hamilton cycles in Cayley digraphs of p-groups to arbitrary vertex-transitive digraphs of prime power order. We give some conditions under which one can obtain Hamilton cycles of vertex-transitive digraphs of prime power order by lifting Hamilton cycles from their quotient graphs. Let Γ be a connected digraph of which the automorphism group contains a finite vertex-transitive subgroup G of prime power order.
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