In this paper graphs for which the chromatic number and the maximal degree are equal to the same number n are considered. It is proved that if such a graph has no \(K_ n\) as a subgraph, then \(| V(G)| \geq 2n-1\) and if G has exactly 2n-1 vertices, then \(n\leq 8\). With the help of a computer the authors showed that there are exactly 13 such graphs having \(2n-1\) vertices. One of these graphs is the smallest counterexample to the Hajos conjecture.