Finite commutative rings with higher genus unit graphs

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface S-g. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.