Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maximizing $H$-colorings of connected graphs with fixed minimum degree   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 $\delta$ maximizes the number of
$H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the
complete bipartite graph $K_{\delta,n-\delta}$ is the unique maximizer. As a
corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$
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 $\delta$ which has more $K_{q}$-colorings (for sufficiently
large $q$ and $n$) than $K_{\delta,n-\delta}$.

36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat