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

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