Counting the number of non-equivalent vertex colorings of a graph

A sharp lower bound on the number of non-equivalent colorings of graphs of order n and maximum degree n− 3

Two vertex colorings of a graph G are equivalent if they induce the same partition of the vertex set into color classes. The graphical Bell number B(G) is the number of non-equivalent vertex colorings of G. We determine a sharp lower bound on B(G) for graphs G of order n and maximum degree n− 3, and we characterize the graphs for which the bound is attained.

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

A sharp lower bound on the number of non-equivalent colorings of graphs of order and maximum degree

Two vertex colorings of a graph G are equivalent if they induce the same partition of the vertex set into color classes. The graphical Bell number B(G) is the number of non-equivalent vertex colorings of G. We determine a sharp lower bound on B(G) for graphs G of order n and maximum degree n− 3, and we characterize the graphs for which the bound is attained.

Counting Colorings of a Regular Graph

At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph Kd,d. In this note we give asymptotic evidence for this conjecture, showing that the number of proper q-colorings admitted by an n-vertex, d-regular...

On Approximately Counting Colorings of Small Degree Graphs

We consider approximate counting of colorings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree ∆ of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by consideri...