  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.
