Ohba’s conjecture for graphs with independence number five

Ohba has conjectured that if G is a k -chromatic graph with at most 2 k + 1 vertices, then the list chromatic number or choosability ch ( G ) of G is equal to its chromatic number χ ( G ) , which is k . It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k -partite graph and each part has at most five vertices.