  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.
