Coloring Colorable Graphs

The Chef has a list of dishes he must prepare and he is able to work on many of them simultaneously (with the help of his employees, of course). For example, he can heat soup, bake a cake, marinate some meat, let some wine aerate, etc. all at the same time since no two of them require a common cooking tool. Unfortunately, his kitchen is dangerously low in its supply of tools and he has no more tha