On certain hyperplane arrangements and colored graphs

We exhibit a one-to-one correspondence between $3$-colored graphs and subarrangements of certain hyperplane arrangements denoted ${\mathcal{J}}_n$, $n \in \mathbb N$\mathrm{.}Wedefinethenotionofcentralityof$3$-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 $n=2,3$.