Arc-coloring of directed hypergraphs and brick-coloring of walls

We define an arc-coloring for a class of directed hypergraphs, in such a way that any two arcs having either intersecting tails or the same head must be colored differently. We investigate the arc-coloring of non-overlapping hypergraphs, namely hypergraphs which can be represented by 2-dimensional adjacency matrices (walls). In particular, coherent walls of degree d - corresponding to interval hypergraphs of degree d - are shown to be optimally colorable in at most 2d - 1 colors. Such property is the first step toward the chromatic classification of coherent walls of fixed degree. As the main result in that vein, we settle the above problem for the degree equal to 2, by providing an enumeration of "essentially distinct" walls in terms of necklaces and beads.