The concept of the \textit{relative fractional packing number} between two graphs $G$ and $H$, initially introduced in arXiv:2307.06155 [math.CO], serves as an upper bound for the ratio of the zero-error Shannon capacity of these graphs. Defined as: \begin{equation*} \sup\limits_{W} \frac{\alpha(G \boxtimes W)}{\alpha(H \boxtimes W)} \end{equation*} where the supremum is computed over all arbitrary graphs and $\boxtimes$ denotes the strong product of graphs. This article delves into various critical theorems regarding the computation of this number. Specifically, we address its NP-hardness and the complexity of approximating it. Furthermore, we develop a conjecture for necessary and sufficient conditions for this number to be less than one. We also validate this conjecture for specific graph families. Additionally, we present miscellaneous concepts and introduce a generalized version of the independence number that gives insights that could significantly contribute to the study of the relative fractional packing number.

翻译：两个图$G$和$H$之间的\textit{相对分数打包数}概念最初在arXiv:2307.06155 [math.CO]中引入，用于作为这些图零误差香农容量比的上界。其定义为：\begin{equation*} \sup\limits_{W} \frac{\alpha(G \boxtimes W)}{\alpha(H \boxtimes W)} \end{equation*}，其中上确界遍历所有任意图，$\boxtimes$表示图的强乘积。本文深入探讨了有关该数计算的若干关键定理。具体而言，我们讨论了其NP难性及其近似计算的复杂性。此外，我们提出了一个关于该数小于1的充要条件的猜想，并验证了该猜想在特定图族中的成立性。同时，我们还介绍了若干杂项概念，并提出了独立数的一个推广版本，其见解可能对相对分数打包数研究具有重要贡献。