Planarity and genus of the n-total graphs of finite commutative rings

Abstract

Let R be a finite commutative ring and let n be a positive integer. The n-total graph τ_n(R) has vertex set R, with distinct vertices x and y adjacent precisely when xⁿ + yⁿ is a zero-divisor. We determine the orientable genus of τ_n(R) for every finite local ring in terms of the order of its maximal ideal, the order q of its residue field, and gcd(n, q - 1). The resulting formula yields a complete planarity criterion and exactly six families of toroidal local ring/exponent pairs. For finite nonlocal rings, we combine a coordinate-zero clique with a residue-fiber adjacency rule to reduce the low-genus problem to a finite collection of products. This gives complete classifications of all nonlocal rings whose n-total graphs are planar or toroidal. We further prove that the least common multiple of the integers q_i - 1, taken over the residue fields of the local factors of R, is a period of the labeled graph and hence of its genus. When n = 1, our results recover the known genus-at-most-one classification for ordinary total graphs.