4-colouring of generalized signed planar graphs

Assume G is a graph and S is a set of permutations of positive integers. An S-signature of G is a pair (D, sigma), where D is an orientation of G and sigma : E(D) -> S is a mapping which assigns to each arc e = (u,v) a permutation sigma(e) in S. We say G is S-k-colourable if for any S-signature (D, sigma) of G, there is a mapping f : V (G) -> [k] such that for each arc e = (u, v) of G, sigma (e)(f (u)) # f (v). The concept of S-k-colourable is a common generalization of many colouring concepts. This paper studies the problem as to which subsets S of S-4, every planar graph is S-4-colourable. We call such a subset S of S-4 a good subset. The Four Colour Theorem is equivalent to saying that S = {id} is good. It was proved by Jin, Wong and Zhu (arXiv:1811.08584) that a subset S containing id is good if and only if S = {id}. In this paper, we prove that, up to conjugation, every good subset of S-4 not containing id is a subset of {(12), (34), (12)(34)}.