Vertex-distinguishing chromatic index of digraphs

Abstract

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 $\chi_{vd}^{\prime}(D)$. In 2016, Li, Bai, He, and Sun conjectured that $\chi_{vd}^{\prime}(D)=k(D)$ 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.