Let G be a simple graph with n vertices. Let k be a positive integer.
Prove the following:
((a)) If G has a proper k-coloring, then G has no subgraph isomorphic to Kk+1.
((b)) If k≥n−2, then the converse to part (a) also holds: If G has no subgraph isomorphic to Kk+1, then G has a proper k-coloring.
((c)) Does the converse to part (a) hold for k<n−2 as well? Specifically, does it hold for n=5 and k=2?