One-dependent colorings of the star graph

Abstract

This paper is concerned with symmetric 1-dependent colorings of the d-ray star graph ${\mathcal{S}}^{\mathit{d}}$ for $\mathit{d}\ge 2$. We compute the critical point of the 1-dependent hard-core processes on ${\mathcal{S}}^{\mathit{d}}$, which gives a lower bound for the number of colors needed for a 1-dependent coloring of ${\mathcal{S}}^{\mathit{d}}$. We provide an explicit construction of a 1-dependent q-coloring for any $\mathit{q}\ge 5$ of the infinite subgraph ${\mathcal{S}}_{(1,1,\infty )}^3$, which is symmetric in the colors and whose restriction to any path is some symmetric 1-dependent q-coloring. We also prove that there is no such coloring of ${\mathcal{S}}_{(1,1,\infty )}^3$ with $\mathit{q}=4$ colors. A list of open problems are presented.