Chromatic Number

Relevant wiki: Graph Coloring and Chromatic Numbers

There need 4 color to color the vertex of the graph.4 is the minimum number of color.Because notice that,there is some triangles in the graph.So,the 3 vertex of the triangle are mutually connected.So,there must need 3 color.Another vertex is connected with three of the vertex of the triangle.So,that vertex must be of differrent color.So,4 color must be needed.

Relevant wiki: Graph Coloring and Chromatic Numbers

The chromatic number is four, because the graph is planar and every planar graph can be colores with 4 colors, but the graph has a sub-clique of size four, and a clique of order n cannot be colored with less than n colors

Note: a planar graph is a graph that can be drawn on a plane without edges intersection

A clique of order n is a graph with n verices where each two vertices are connected