I need to show, that the following graph is 5-colored or even better 4-colored. But I the only thing I got so far is 6-colored. The original graph is on the left, on the right is the result I have so far.
The problem is, that the (b) vertex alone has connections to 5 other vertices, so I need at least 6 colors. Am I right or am I missing something?
You need to give different colors for vertices that are adjacent. So the vertices $e, d, c, f$ can have same color. You can see that two more colors will be needed to color all the vertices of this graph properly. So this is a $3$ colorable graph!