In this paper, we use a topological quantum field theory (TQFT) to define families of new homology theories of a $2$-dimensional CW complex of a smooth closed surface. The dimensions of these homology groups can be used to count the number of ways that each face of the CW complex can be colored with one of $n$ colors so that no two adjacent faces have the same color. We use these homologies to define new invariants of graphs, give new characterizations of well-known polynomial invariants of graphs, and rephrase and offer new approaches to famous conjectures about graph coloring. In particular, we show that the TQFT has the potential to generate $4$-face colorings of a bridgeless planar graph, leading to a constructive approach to the four color theorem. The TQFT has ramifications for the study of smooth surfaces and provides examples of new types of Frobenius algebras.