Maximizing the number of q-colorings

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

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...

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 δ maximizes the number of H-colorings? We show that for non-regular H and sufficiently large n, the complete bipartite graph Kδ,n−δ is the unique maxim...

Quandle colorings of knots and applications.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the...

Fundamentally Different m-Colorings of Pretzel Knots

We show that the number of fundamentally different m-colorings of a knot K depends only on the m-nullity of K, and develop a formula for the number of such colorings. We also determine the m-colorability and m-nullity of any (p, q, r) pretzel knot, and therefore determine the number of fundamentally different m-colorings for any (p, q, r)