A simple-intersection graph of a ring approach to solving coloring optimization problems

In this paper, we introduce a modified version of the simple-intersection graph for semisimple rings, applied to a ring $R$ with unity. The findings from this modified version are subsequently utilized to solve several coloring optimization problems.  We demonstrate how the clique number of the simple-intersection graph can be used to determine the maximum number  of possibilities that can be selected from a set of $n$ colors without replacement or order, subject to the constraint that  any pair shares only one common color. We also show how the domination number can be used to determine the  minimum number of possibilities that can be selected, such that any other possibility shares one color with  at least one of the selected possibilities, is $n-1$.