What is the minimum number of edges in a -chromatic connected graph of order ?

I have read somewhere that this number is equal to

I was thinking about using , the complete graph with vertices. We know that the number of edges in is equal to .

Also, I wanted to use the fact that if the chromatic number of a graph , is equal to , and if a vertex has degree strictly less than , then the chromatic number of the graph ( together with and its edges) is also equal to .

I am not sure how start, I need help organizing my ideas.