The rings considered in this article are commutative with identity 1 = 0. Recall that the unit graph of a ring R is a simple undirected graph whose vertex set is the set of all elements of the ring R and two distinct vertices x, y are adjacent in this graph if and only if x + y ∈ U(R) where U(R) is the set of unit elements of ring R. We denote this graph by UG(R). In this article we classified local ring R such that UG(R) is planar.

The rings considered in this article are commutative with identity 1 = 0. Recall that the unit graph of a ring R is a simple undirected graph whose vertex set is the set of all elements of the ring R and two distinct vertices x, y are adjacent in this graph if and only if x + y ∈ U(R) where U(R) is the set of all unit elements of ring R. We denote this graph by UG(R). In this article we classified rings R with |Max(R)| = 2 such that UG(R) is planar.

The rings considered in this article are commutative with identity 1 = 0. Recall that the unit graph of a ring R is a simple undirected graph whose vertex set is the set of all elements of the ring R and two distinct vertices x, y are adjacent in this graph if and only if x + y ∈ U(R) where U(R) is the set of all unit elements of ring R. We denote this graph by UG(R). In this article we classified rings R with |Max(R)| ≥ 3 such that UG(R) is planar.

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The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of R, denoted by C (R) is an undirected simple graph whose vertex set is the set of all proper ideals I of R such that I ⊆ J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2 are joined by an edge in C (R) if and only if I1 + I2 = R.