Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident to any its neighbour. We provide an infinite family of examples of graphs $G$ with $\chi'_\subset(G)\geq (1+\frac{1}{\delta-1})\Delta$, where $\Delta$ is the maximum degree of $G$, and we conjecture that $\chi'_\subset(G)\leq \lceil(1+\frac{1}{\delta-1})\Delta\rceil$ for every connected graph with $\delta\geq 2$ which is not isomorphic to $C_5$. The equality here is attained e.g. for the family of complete bipartite graphs. Using a probabilistic argument we support this conjecture by proving that for any fixed $\delta\ge2$, $\chi'_\subset(G) \le (1+\frac{4}{\delta})\Delta (1+o(1))$ (for $\Delta\to\infty$), what implies that $\chi'_\subset(G) \le (1+\frac{4}{\delta-1})\Delta$ for $\Delta$ large enough.

中文翻译：

---

关于图的包含色指数

令 $\chi'_\subset(G)$ 为以最小度数 $\delta\geq 2$ 为图 $G$ 的边正确着色所需的最少颜色数，以便与任何顶点相交的颜色集不包含在与其任何邻居相关的一组颜色中。我们提供了一系列图形 $G$ 和 $\chi'_\subset(G)\geq (1+\frac{1}{\delta-1})\Delta$，其中 $\Delta$ 是$G$ 的最大度数，我们推测每个连通图的 $\chi'_\subset(G)\leq \lceil(1+\frac{1}{\delta-1})\Delta\rceil$ $\delta\geq 2$ 与 $C_5$ 不同构。例如，对于完全二部图族，这里的等式是实现的。使用概率论点，我们通过证明对于任何固定的 $\delta\ge2$，$\chi' 来支持这个猜想