标题： 边开放包装：进一步的表征 显示英文标题

标题： Edge open packing: further characterizations

摘要： 设 $G=(V, E)$ 是一个图，其中 $V(G)$ 和 $E(G)$ 分别是顶点集和边集。 在一个图$G$中，两条边$e_1, e_2\in E(G)$被称为\emph{公共边} $e\neq e_1, e_2$ ，如果$e$将$e_1$的一个端点连接到$e_2$的一个端点在$G$中。 一个子集$D\subseteq E(G)$在$G$中被称为\emph{边开放打包集}，如果$D$中的任何两条边在$G$中不共享一条公共边，并且这样的集合在$G$中的最大大小被称为\emph{边开放包装数}，用$\rho_{e}^o(G)$表示。 在介绍性论文（Chelladurai 等 (2022)）中，提供了$\rho_{e}^o(G)=1, 2$的必要和充分条件，并表征了具有$\rho_{e}^o(G)\in \{m-2, m-1, m\}$的图$G$，其中$m$是图$G$的边数。 在本文中，我们进一步表征了图$G$。 首先，我们证明$\rho_{e}^o(G)=t$的必要且充分条件，对于任何整数$t\geq 3$。 最后，我们描述具有$\rho_{e}^o(G)=m-3$的图。 显示英文摘要

摘要： Let $G=(V, E)$ be a graph where $V(G)$ and $E(G)$ are the vertex and edge sets, respectively. In a graph $G$, two edges $e_1, e_2\in E(G)$ are said to have \emph{common edge} $e\neq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ in $G$. A subset $D\subseteq E(G)$ is called an \emph{edge open packing set} in $G$ if no two edges in $D$ share a common edge in $G$, and the largest size of such a set in $G$ is known as \emph{edge open packing number}, represented by $\rho_{e}^o(G)$. In the introductory paper (Chelladurai et al. (2022)), necessary and sufficient conditions for $\rho_{e}^o(G)=1, 2$ were provided, and the graphs $G$ with $\rho_{e}^o(G)\in \{m-2, m-1, m\}$ were characterized, where $m$ is the number of edges of $G$. In this paper, we further characterize the graphs $G$. First, we show necessary and sufficient conditions for $\rho_{e}^o(G)=t$, for any integer $t\geq 3$. Finally, we characterize the graphs with $\rho_{e}^o(G)=m-3$.