Maximizing the number of x-colorings of 4-chromatic graphs

Abstract Let C 4 ( n ) be the family of all connected 4 -chromatic graphs of order n . Given an integer x ≥ 4 , we consider the problem of finding the maximum number of x -colorings of a graph in C 4 ( n ) . It was conjectured that the maximum number of x -colorings is equal to ( x ) ↓ 4 ( x − 1 ) n − 4 and the extremal graphs are those which have clique number 4 and size n + 2 . In this article, we reduce this problem to a finite family of graphs. We show that there exists a finite family F of connected 4 -chromatic graphs such that if the number of x -colorings of every graph G in F is less than ( x ) ↓ 4 ( x − 1 ) | V ( G ) | − 4 then the conjecture holds to be true.

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