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 ${\mathcal{J}}_n$, $n{\in}{\mathbb{N}}$. 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.