First a general remark. Not everybody studied Brooks' theorem and related concepts in college, but most people in this subforum know about graphs and coloring. If it is reasonable to think that the cost of learning about the concepts from your particular problem is not too high, and if you think you can benefit from the "common mathematical knowledge" of many smart(er than me) people on this forum, then it may be worth giving the definitions of the concepts involved and maybe giving some standard online reference. Of course, if it seems that "common knowledge" is not sufficient and one has to be an expert in those particular concepts, then it probably does not make sense to explain a lot, waiting for replies from people who know what you are talking about.

The upper bound given by the Brooks' theorem is the maximum vertex degree. It's relatively easy to see from the definitions that in Peterson's graph every vertex has degree 3, and in every vertex has degree . The 3-coloring of Peterson's graph is shown in Wikipedia. To prove that it is not 2-colorable, try to come up with a 2-coloring making only steps that are necessary (i.e., do not involve guessing the color of some vertex), and you will see that this is impossible.

The graph is bipartite as described in Wikipedia, and it's also relatively easy to see why. Therefore, its chromatic number is 2.

Is it possible that I misunderstood your question and that you were asking not why the numbers are what they are, but why the upper bound is tight for Peterson's graph and not tight for ?