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.

翻译：考虑有向图的边着色，要求边$v_1 v_2$与$v_2 v_3$具有不同颜色。此类着色通过边颜色集合诱导出顶点着色，其中图中边$v_1 v_2$的存在意味着$v_1$的集合颜色包含某个不在$v_2$集合颜色中的元素，反之亦然。我们将此推广至$k$（顶点）路径的着色，其定义为：若两条路径分别是某公共$(k+1)$-路径的前缀$c_1$与后缀$c_2$，则它们必须具有不同颜色。进一步地，颜色可属于偏序集$P$，其中$c_1$与$c_2$须满足$c_1 \not\leq c_2$。该集合构造将$P$中由$k$-路径颜色集合生成的**下序理想**进行推广；这些序理想通过包含关系形成偏序。我们最终证明：$k$-路径的$P$着色存在，当且仅当存在顶点着色方案，该方案通过将Birkhoff算子$A$（将偏序集映射至其下序理想构成的偏序集）在$P$上迭代$k-1$次所得的代数结构$A$实现。因此，有向图着色指标问题被推广并归结为顶点的偏序集着色问题。本研究借鉴了Cole与Vishkin在确定性抛币问题中的思路与成果，以及Becker与Simon在$(n-2)$-立方体子集顶点覆盖问题中的研究动机。