Existentially closed graphs via permutation polynomials over finite fields

For a positive integer n, a graph is n-existentially closed or n-e.c. if we can extend all n-subsets of vertices in all possible ways. It is known that almost all finite graphs are n-e.c. Despite this result, until recently, only few explicit examples of n-e.c. graphs are known for n>2. In this paper, we construct explicitly a 4-e.c. graph via a linear map over finite fields. We also study the colored version of existentially closed graphs and construct explicitly many (3,t)-e.c. graphs via permutation polynomials and multiplicative groups over finite fields.