Maximizing H‐Colorings of Connected Graphs with Fixed Minimum Degree

Abstract

For graphs G and H, an H‐coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H, which connected n‐vertex graph with minimum degree δ maximizes the number of H‐colorings? We show that for nonregular H and sufficiently large n, the complete bipartite graph Kδ,n−δ is the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph Kk,n−k is the unique k‐connected graph that maximizes the number of H‐colorings among all k‐connected graphs. Finally, we show that this conclusion does not hold for all regular H by exhibiting a connected n‐vertex graph with minimum degree δ that has more Kq‐colorings (for sufficiently large q and n) than Kδ,n−δ.

Maximising ‐colourings of graphs

“…In this paper, we will extend this result by showing that it holds for all ≥ δ 3 and for all graphs H . This answers a question posed by Engbers [5]. In the case where δ = 2 and H is any graph, Engbers [4] showed that the number of H -colourings is maximised by one of K K , n n 2, −2 3 3 , or K n 4 2,2 (depending on the structure of H ).…”

“…We have also considered the more general question which was asked by Engbers : what happens if we consider all graphs on vertices with minimum degree , rather than just those which are connected? We will look at the case where is fixed and .…”

“…In this paper, we consider the class of all graphs with minimum degree at least . This class was studied by Engbers who raised a number of questions and conjectures. We will answer two of these and provide a partial answer to a third.…”

“…Throughout this paper, will be a simple graph without loops. We will adopt the same convention for vertex degrees as Engbers : for any vertex , we define . In particular, adding a loop to a vertex in increases the degree by 1.…”

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

“…See [17,19,20,25] for discussions on the analogous problem of maximizing the number of homomorphisms into a fixed H. The following upper bounds on m(G) and pm(G) have a curious semblance to Theorems 1.1 and 1.2. The quantity m(G) for matchings can be viewed as analogous to i(G) for independent sets.…”