When Is the Complement of the Comaximal Graph of a Commutative Ring Planar?

Let be a commutative ring with identity. In this paper we classify rings such that the complement of comaximal graph of is planar. We also consider the subgraph of the complement of comaximal graph of induced on the set of all nonunits of with the property that each element of is not in the Jacobson radical of and classify rings such that this subgraph is planar. 1. Introduction All rings considered in this paper are commutative with identity . The graphs considered here are undirected and simple. Sharma and Bhatwadekar in [1] introduced a graph on a commutative ring , whose vertices are the elements of and two distinct vertices and are adjacent if and only if . In [1, Theorem 2.3], the authors showed that if and only if the ring is finite and in this case , where and denote, respectively, the number of maximal ideals of and number of units of . In [2], Maimani et al. named the graph studied by Sharma and Bhatwadekar as the comaximal graph of . Also in [2], the authors studied the subgraphs , and , where is the subgraph of induced on the set of units of , is the subgraph of induced on the set of nonunits of , and is the subgraph of induced on the set of nonunits of which are not in = the Jacobson radical of . Several other researchers also investigated the comaximal graphs of rings [3–5]. In [6], Gaur and Sharma studied the graph whose vertices are the elements of a ring and two distinct vertices and are adjacent if and only if there exists a maximal ideal of containing both and . They called this graph the maximal graph of the ring . Let be a simple graph. Recall from [7, Definition 1.1.13] that the complement of is defined by taking and two distinct vertices and are adjacent in if and only if they are not adjacent in . Thus the maximal graph of a ring studied in [6] is the complement of the comaximal graph of a ring studied by Sharma and Bhatwadekar in [1]. For any set denotes the cardinality of . For any ring , and denote, respectively, the set of units of and the set of nonunits of . We denote by and the set of all maximal ideals of by Max( ). For the sake of convenience, for any ring with at least two maximal ideals we denote by . For any , denotes the ring of integers modulo . For any prime number and , denotes the finite field with exactly elements. We first recall the following definitions and results from graph theory. A graph is said to be complete if every pair of distinct vertices of are adjacent in . A complete graph on vertices is denoted by [7, Definition 1.1.11]. A graph is said to be bipartite if the vertex set can be partitioned into