Assume $G$ is a graph and $S$ is a set of permutations of integers. An $S$-labelling of $G$ is a pair $(D,\sigma )$, where $D$ is an orientation of $G$ and $\sigma :E\left(D\right)\to S$ is a mapping which assigns to each arc $e$ of $D$ a permutation ${\sigma }_e\in S$. A proper $k$-colouring of $(D,\sigma )$ is a mapping $f:V\left(G\right)\to \left[k\right]=\{1,2,\dots ,k\}$ such that ${\sigma }_e\left(f\left(x\right)\right)\ne f\left(y\right)$ for each arc $e=(x,y)$. We say $G$ is $S$-$k$-colourable if any $S$-labelling $(D,\sigma )$ of $G$ has a proper $k$-colouring. The concept of $S$-$k$-colouring is a common generalization of many colouring concepts, including $k$-colouring, signed $k$-colouring, signed $Z_k$-colouring, DP-$k$-colouring, group colouring and colouring of gain graphs. We are interested in the problem as for which subset $S$ of $S_4$, every planar graph is $S$-4-colourable. We call such a subset $S$ a good subset. The famous Four Colour Theorem is equivalent to say that $S=\left\{id\right\}$ is good. A result of Král, Pangrác and Voss is equivalent to say that $S=\{id,\left(1234\right),\left(13\right)\left(24\right)\}$ and $S=\{id,\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\}$ are not good. These results are strengthened by a very recent result of Kardoš and Narboni, which implies that $S=\{id,\left(12\right)\left(34\right)\}$ is not good and another very recent result of Zhu which implies that $S=\{id,\left(12\right)\}$ is not good. In this paper we prove if $S$ is a subset of $S_4$ containing $id$, then $S$ is good if and only if $S=\left\{id\right\}$.

承担 $G$ 是图 $\mathrm{小号}$是一组整数的排列。一个$\mathrm{小号}$-标签 $G$ 是一对 $（d，\sigma ）$，在哪里 $d$ 是一个方向 $G$ 和 $\sigma ：Ë（d）\to \mathrm{小号}$ 是分配给每个弧的映射 $Ë$ 的 $d$ 排列 ${\sigma }_{Ë}\in \mathrm{小号}$。正确的$ķ$-着色 $（d，\sigma ）$ 是一个映射 $F：V（G）\to \left[ķ\right]=\{\mathrm{1个}，2，\dots ，ķ\}$ 这样 ${\sigma }_{Ë}（F（X））\ne F（ÿ）$ 对于每个弧 $Ë=（X，ÿ）$。我们说$G$ 是 $\mathrm{小号}$--$ķ$-如有色 $\mathrm{小号}$-标签 $（d，\sigma ）$ 的 $G$ 有一个适当的 $ķ$-染色。概念$\mathrm{小号}$--$ķ$着色是许多着色概念的通用概括，包括 $ķ$-着色，签名 $ķ$-着色，签名 ${ž}_{ķ}$-着色，DP-$ķ$-着色，组着色和增益图着色。我们对哪个子集的问题感兴趣$\mathrm{小号}$ 的 ${\mathrm{小号}}_4$，每个平面图是 $\mathrm{小号}$-4-有色。我们称这样的子集$\mathrm{小号}$一个很好的子集。著名的四色定理等于说$\mathrm{小号}=\left\{\mathrm{一世}d\right\}$很好。Král，Pangrác和Voss的结果等于说$\mathrm{小号}=\{\mathrm{一世}d，（1234），（13）（24）\}$ 和 $\mathrm{小号}=\{\mathrm{一世}d，（12）（34），（13）（24），（14）（23）\}$不好 Kardoš和Narboni的最新结果进一步加强了这些结果，这表明$\mathrm{小号}=\{\mathrm{一世}d，（12）（34）\}$ 是不好的，朱的另一个最近的结果暗示 $\mathrm{小号}=\{\mathrm{一世}d，（12）\}$不好 在本文中，我们证明$\mathrm{小号}$ 是...的子集 ${\mathrm{小号}}_4$ 包含 $\mathrm{一世}d$， 然后 $\mathrm{小号}$ 当且仅当 $\mathrm{小号}=\left\{\mathrm{一世}d\right\}$。