Exploring Graph Embedding Strategies for Co-Intersection Ideals in Commutative Rings

This research article examines the co-intersection graph of a commutative ring 𝒮. The co-intersection graph Ω(𝒮) is defined as a simple graph where the vertices correspond to the non-trivial ideals of 𝒮, and two distinct vertices I and J are adjacent if and only if I+J≠𝒮. In this study, we first classify the Artinian rings 𝒮 for which Ω(𝒮) is isomorphic to certain basic graphs, such as a unicycle, a split graph, or a threshold graph. Subsequently, we investigate the genus of Ω(𝒮) and identify the rings 𝒮 for which Ω(𝒮) represents a double-toroidal graph. Furthermore, we analyze the crosscap of Ω(𝒮) and determine the rings 𝒮 for which Ω(𝒮) corresponds to the projective plane or the Klein bottle. Lastly, we compute the book thickness of Ω(𝒮) for cases where the genus is at most one.