有向图的顶点可区别色指数
Vertex-distinguishing chromatic index of digraphs
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中文总结 AI 辅助
该研究聚焦有向图顶点可区别色指数问题，证实了Li等人2016年提出的“至多含一个源点和一个汇点的有向图的顶点可区别色指数等于由出度、入度序列确定的自然下界”的猜想。
中文摘要 AI 辅助
设$D$为一个有向图。在本注记中，$D$的**弧着色**是指为$D$的各条弧分配颜色，使得任意两条具有公共尾的弧颜色不同，且任意两条具有公共头的弧颜色不同。在该着色规则下，每个顶点$v$对应一个**出色集合**和一个**入色集合**，分别由以$v$为尾的弧的颜色、以$v$为头的弧的颜色构成。若$D$的一种弧着色满足任意两个不同顶点的出色集合不同、入色集合也不同，则称该弧着色是**顶点可区别**的。$D$的顶点可区别弧着色所需的最少颜色数，称为$D$的顶点可区别色指数，记为$χ_{vd}^{\'}(D)$。2016年，Li、Bai、He与Sun提出猜想：对于至多含一个源点、至多含一个汇点的任意有向图$D$，有$χ_{vd}^{\'}(D)=k(D)$，其中$k(D)$是由$D$的出度序列与入度序列确定的自然下界。我们证实了该猜想。
英文摘要
Let $D$ be a digraph. In this note, an \emph{arc coloring} of $D$ is an assignment of colors to the arcs of $D$ such that no two arcs with a common tail receive the same color and no two arcs with a common head receive the same color. Under such a coloring, each vertex $v$ is associated with an \emph{out-color set} and an \emph{in-color set}, consisting of the colors assigned to the arcs with tail $v$ and to the arcs with head $v$, respectively. An arc coloring of $D$ is \emph{vertex-distinguishing} if any two distinct vertices have different out-color sets and different in-color sets. The minimum number of colors required for a vertex-distinguishing arc coloring of $D$ is called the \emph{vertex-distinguishing chromatic index} of $D$, denoted $χ_{vd}^{\prime}(D)$. In 2016, Li, Bai, He, and Sun conjectured that $χ_{vd}^{\prime}(D)=k(D)$ for any digraph $D$ with at most one source and at most one sink, where $k(D)$ is a natural lower bound determined by the outdegree and indegree sequences of $D$. We confirm this conjecture.
