Abstract by Jennifer Canizales
Coloring the Map of Utah
The “four color theorem” asserts that any planar graph is four colorableーmeaning four colors are sufficient to properly color any planar graph. Let p(G,r) be the number of ways to color a graph G with r colors. It’s known that p(G,r) is a monic polynomial of degree |V(G)| with integer coefficients and is called the chromatic polynomial. It is surprising that no one has computed this polynomial for real world graphs, like the graph of the map of the counties of Utah. So, in this project, the four of us have done that.
In the process of finding the chromatic polynomial for the map of Utah I discovered a pattern among what we call "interlocking wheels." Here I will present the theorem and proof for the chormatic polynomial of two interlocking wheels of any size and how we have used this theorem to find the chromatic polynomial for the graph of the counties of Utah.