A Theory for Coloring Walks in a Digraph

Consider edge colorings of digraphs where edges $v_1 v_2$ and $v_2 v_3$ have different colors. This coloring induces a vertex coloring by sets of edge colors, in which edge $v_1 v_2$ in the graph implies that the set color of $v_1$ contains an element not in the set color of $v_2$, and conversely. We generalize to colorings of $k$(vertex)-walks, defined so two walks have different colors if one is the prefix $c_1$ and the other is the suffix $c_2$ of a common $(k+1)$-walk. Further, the colors can belong to a poset $P$ where $c_1$, $c_2$ must satisfy $c_1 \not\leq c_2$. This set construction generalizes the lower order ideal in $P$ from a set of $k$-walk colors; these order ideals are partially ordered by containment. We conclude that a $P$ coloring of $k$-walks exists iff there is a vertex coloring by $A$ iterated $k-1$ times on $P$, where Birkhoff's $A$ maps a poset to its poset of lower order ideals. Thus the directed chromatic index problem is generalized and reduced to poset coloring of vertices. This work uses ideas, results and motivations due to Cole and Vishkin on deterministic coin tossing and Becker and Simon on vertex covers for subsets of $(n-2)$-cubes.