Coloring of graphs associated with commutative rings

For a commutative ring R , even elements in R as vertices and two distinct vertices x i , y j ∈ R are adjacent iff x i y j = 0 or y j x i = 0 , then the graph is known as an even prime graph. In compressed even prime graph, whose vertex set is the set of all equivalence classes of even elements in R , the equivalence classes elements are denoted by [ β ] or β ¯ and two distinct equivalence classes [ β i ] and [ β j ] are adjacent iff [ β i ] [ β j ] = 0 , graph is denoted by E P G E ( Z n ) . This paper delves into discussions regarding the chromatic number and clique number of such graphs across various families. Additionally, we discuss the chromatic number of the prime graph associated with the commutative ring R .