Colorings of the d-regular infinite tree

Strong Colorings Yield Κ-bounded Spaces with Discretely Untouchable Points

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklóssy and the first author, we show that this statement fails for countably compact regular spaces, and even for ω-bounded regular spaces. In fact, there are κ-bounded counterexamples for every infinite cardinal κ. The proof makes ...

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

Extremal H-colorings of trees and 2-connected graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several res...

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...

Sharp Phase Transition in the Random Stirring Model on Trees

We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d0 such that, for any d ≥ d0, the set of parameter values at which the random stirring model on the rooted regular tree with offspring degree d almost surely contains an infinite cycle consists of a semi-infinite interva...