Abstract

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