Hello ladies and gentlemen.
It is an undirected graph, it is $3$-regular and it's chromatic number is $3$. Proof: There is a circle with $5$ nodes (the outside pentagon), a graph with a graph that contains a circle with an odd number of nodes is at least $3$. In the picture I gave a coloring with $3$ colors, so it is at most $3$ as well. The result is that the chromatic number is $3$.
The question is: What is the chromatic number of the complement of petersen graph? We can deduce that it will be $6$-regular, and the pentagram on the inside of petersen graph will be a pentagon in the complement, so the complement has chromatic number of at least $3$. I don't know where to go from here without actually color the graph and hope that I can do it with $3$ colors.
Important Edit: I found a clique of $4$ nodes in the complement, so the chromatic number is at least $4$.