  Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge
$uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and
$e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of
positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $\sigma:
E(G) \to S$ with $\sigma (x,y) = (\sigma (y,x))^{-1}$, there is a mapping $f:
V(G) \to [k]=\{1,2, \ldots, k\}$ such that for each arc $e=(x,y)$,
$\sigma_e(f(x)) \ne f(y)$.
  The concept of $S$-$k$-colouring is a common generalization of many colouring
concepts, including $k$-colouring, signed $k$-colouring defined by
M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera, signed $k$-colouring defined by
Kang and Steffen, correspondence $k$-colouring defined by Dvo\v{r}\'{a}k and
Postle, and group colouring defined by Jaeger, Linial, Payan and Tarsi.
  We are interested in the problem as for which subset $S$ of $S_4$, every
planar graph is $S$-colourable. Such a subset $S$ is called good. The famous
four colour theorem is equivalent to say that $S=\{id\}$ is good. There are two
conjectures on signed graph colouring, one is equivalent to $S=\{id,
(12)(34)\}$ be good and the other is equivalent to $S=\{id, (12)\}$ be good. We
say two subsets $S$ and $S'$ of $S_k$ are conjugate if there is a permutation
$\pi \in S_k$ such that $S'= \{\pi \sigma\pi^{-1}: \sigma \in S\}$. This paper
proves that if $S$ is a good subset of $S_4$ containing $id$, then $S$ is
conjugate to a subset of $\{id, (12), (34), (12)(34)\}$. However, it remains an
open problem if there is any good subset $S$ which contains $id$ and has
cardinality $|S| \ge 2$. We also prove that $S=\{(12),(13),(23),(123),(132)\}$
is not good.
