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If the face\mbox{-}cycles at all the vertices in a map are of same type thenthe map is called semi\mbox{-}equivelar. A map is called minimal if the numberof vertices is minimal. We know the bounds of number of vertex orbits ofsemi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} hasproved that every semi-equivelar toroidal map has a vertex-transitive cover. Inthis article, we prove that if a semi-equivelar map is $k$ orbital then it hasa finite index $m$-orbital minimal cover for $m \le k$. We also show theexistence and classification of $n$-sheeted covers of semi-equivelar toroidalmaps for each $n \in \mathbb{N}$.

Semi-equivelar toroidal maps and their vertex covers