Normality of one-matching semi-Cayley graphs over finite abelian groups with maximum degree three

A graph $\Gamma$ is said to be a semi-Cayley graph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. We say that $\Gamma$ is normal if $G$ is a normal subgroup of ${\rm Aut}(\Gamma)$. We prove that every connected intransitive one-matching semi-Cayley graph, with maximum degree three, over a finite abelian group is normal and characterize all such non-normal graphs.
Comment: 10 pages