Vertex Coloring : How do we know that this is an optimal coloring?

Question

I was solving this question related to vertex colorings. In the solution part of the question it says that:

"The coloring is optimal because the graph contains the complete graph (clique) K4."

Also in another question, the same explanation goes:

"The coloring is optimal because the vertices 1 to 5 form a complete subgraph K5."

Why a vertex coloring (or edge coloring too? ) has to be optimal since it contains a complete graph? I have gone through all my lecture notes and slides but there is no such an information.

Thanks for your help, in advance!

Answer 1

These arguments work in a quite specific case. They show that the first graph cannot have a colouring with fewer than 4 colours, and the second graph cannot have a colouring with fewer than 5 colours. So any 4-colouring of the first graph is optimal, and any 5-colouring of the second graph is optimal.

A graph that contains K₄ can never be coloured with fewer than 4 colours, because that subgraphs forbids it. K₄ has four nodes all connected to each other, so each of them has to have a different colour, so you need 4 colours just for that subgraph. If that subgraph already needs 4 colours, the entire graph definitely does - though perhaps it needs more (in that case you'd need a different kind of proof of optimality).

An other way to look at the graph colouring constraints, perhaps helpful in this case, is to interpret every clique as an all-different constraint.