Coloring Graphs with Graphs: a Survey

Many problems in and areas of graph theory can be thought of in terms of graph homomorphisms. For graphs G and H, a homomorphism from G to H is a function φ : V (G)→ V (H) such that if xy ∈ E(G) then φ(x)φ(y) ∈ E(H). (See [1] for a comprehensive introduction.) Let Hom(G,H) be the set of homomorphisms from G to H. In this survey, we will be interested in counting homomorphisms from G to H and so we let hom(G,H) = |Hom(G,H)|. When G is small relative to H, then homomorphisms from G to H roughly correspond to embeddings of G inside of H. We will focus on the case when G is large and H is small. In this case, we can think of a homomorphism from G to H as a labeling of the vertices of G with the vertices of H. Thus, we sometimes refer to a homomorphism from G to H as an H-coloring of G. Since the labeling corresponds to a homomorphism, vertices in G that are adjacent can only receive labels of vertices that are adjacent in H. We will want to allow that adjacent vertices in G receive the same label, and so permit the image graph H to have loops. We will assume throughout this survey that the pre-image graph G is simple, and so has no loops or multiple edges. Let us consider some examples of H-colorings of graphs for various H.