Let $G=(V,E)$ be a graph and $c:(V\cup E)\to C$ be a proper total colouring of G, where C is a set of colours. We call c inclusion-free if for each vertex, the set of colours appearing on the vertex and the incident edges is not a subset of the respective sets of its neighbours. With a probabilistic argument we show that the minimum number of colours for inclusion-free total colouring, denoted by ${\chi }_{\subset }^{″}(G)$, is bounded from above by $\mathrm{\Delta }+150\log \mathrm{\Delta }$ for any graph G with large enough maximum degree Δ. Then we prove that ${\chi }_{\subset }^{″}(G)\le 6$ for any subcubic graph G, which meets the bound for adjacent vertex distinguishing total colouring.

中文翻译：

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包含总色数

让 $G=(伏,乙)$ 是一个图形和 $C：(伏\cup 乙)\to C$是G的适当总着色，其中C是一组颜色。如果对于每个顶点，出现在顶点和入射边上的颜色集不是其相邻的相应集的子集，我们称c为无包含。通过概率论，我们证明了无夹杂总着色的最小颜色数，表示为${\chi }_{\subset }^{”}(G)$, 由上界 $\mathrm{\Delta }+150\mathrm{日志}\mathrm{\Delta }$对于任何具有足够大的最大度 Δ 的图G。然后我们证明${\chi }_{\subset }^{”}(G)\le 6$对于任何亚立方图G，它满足相邻顶点区分总着色的界限。