Bounding problem 2

Here is the graph again for reference:


The chromatic number is 4.

The following picture shows that four colors is sufficient to color the graph:

To show that four colors is necessary, notice that vertices a, c, e, d, g form a 5-cycle. Since h is directly connected to all of them, the graph contains a 5-wheel. Odd wheels are well-known to have chromatic number 4.