One-dependent colorings of the star graph

TL;DR

This paper investigates symmetric 1-dependent colorings of star graphs, determining the minimum number of colors needed and providing explicit constructions for certain cases, while proving impossibility results for others.

Contribution

It computes the critical point for 1-dependent hard-core processes on star graphs and constructs explicit 1-dependent colorings for the infinite subgraph with at least 5 colors.

Findings

Critical point for 1-dependent hard-core processes computed

Explicit 1-dependent q-coloring constructed for q ≥ 5

Proved no such coloring exists with q = 4

Abstract

This paper is concerned with symmetric $1$-dependent colorings of the $d$-ray star graph $\mathscr{S}^d$ for $d \ge 2$. We compute the critical point of the $1$-dependent hard-core processes on $\mathscr{S}^d$, which gives a lower bound for the number of colors needed for a $1$-dependent coloring of $\mathscr{S}^d$. We provide an explicit construction of a $1$-dependent $q$-coloring for any $q \ge 5$ of the infinite subgraph $\mathscr{S}^3_{(1,1,\infty)}$, 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 $\mathscr{S}^3_{(1,1,\infty)}$ with $q = 4$ colors. A list of open problems are presented.