# Understanding CLIQUE structure

I am working on the following problem:

Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex is connected to every other vertex. Recall also that a clique is a complete subset of some graph. The graph coloring problem consists of assigning a color to each of the vertices of a graph such that adjacent vertices have different colors and the total number of colors used is minimized. We define the chromatic number of a graph G to be this minimum number of colors required to color graph G. Prove that the chromatic number of a graph G is no less than the size of a maximal clique of G.

So far, I have been thinking about the problem and came up with the following:

• In a clique, every vertex is adjacent to every other vertex
• Therefore, since the problem states adjacent verteces always have different colors, the number of colors needed would be the number of verteces in the clique
• Thus, the chromatic number of any graph G has to be at least the maximal size of the clique, per the definition of a clique.

Can someone verify if my understanding is correct? Or am I glaringly not understanding a clique?

Your understanding is right. So in particular, for any graph $$G$$ with a clique of size $$k$$, you will need at least $$k$$ colors to properly color it. In case this is not obvious, you can think about it via a contradiction. So assume $$G$$ contains a clique of size $$k$$, and you have a proper coloring with less than $$k$$ colors. It follows that the clique contains at least two vertices with the same color. By definition of a clique, these two vertices are adjacent. But this is a contradiction to the coloring being proper.