  We exhibit a one-to-one correspondence between $3$-colored graphs and
subarrangements of certain hyperplane arrangements denoted $\mathcal J_n$, $n
\in \mathbb N$. We define the notion of centrality of $3$-colored graphs which
corresponds to the centrality of hyperplane arrangements. Via the
correspondence, the characteristic polynomial $\chi_{\mathcal J_n}$ of
$\mathcal J_n$ can be expressed in terms of the number of central $3$-colored
graphs, and we compute $\chi_{\mathcal J_n}$ for $n = 2, 3$.
