A Theory for Coloring Walks in a Digraph

Consider edge colorings of directed graphs where edges of the form v1v2 and v2v3 must have different colors. Here, v1 ≠ v2 , v2 ≠ v3 but v1 = v3 is possible. It is known that this coloring induces a vertex coloring by sets of edge colors, in which edge v1v2 in the graph implies that the set color of v1 contains an element not in the set color of v2; conversely, each such set coloring of vertices induces one or more edge colorings. We show that these relationships generalize to colorings of of k(vertex)-walks in which two k-walks have different colors if one is the prefix and the other is the suffix of a common (k+1)-walk. For full generality the colors belong to a partially ordered set P; and the prefix color c1 and the suffix color c2 must satisfy c1 ≤| c2 . The set color construction generalizes to generating the lower order ideal in P from a set of k-walk colors; these order ideals (antichains in P, equivalently) are partially ordered by containment. We conclude that a P coloring of k-walks exists if and only if there is a vertex coloring by Ak−1(P), where A is the operator that maps a poset to its poset of lower order ideals, due to Birkhoff. In the case when the graph G is symmetric, this condition means that the largest antichain size (Dilworth