Colouring of [formula omitted]-labelled planar graphs

Abstract

Assume G is a graph and S is a set of permutations of integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say G is S-k-colourable if any S-labelling (D,σ) 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 Zk-colouring, DP-k-colouring, group colouring and colouring of gain graphs. We are interested in the problem as for which subset S of S4, 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={id} is good. A result of Král, Pangrác and Voss is equivalent to say that S={id,(1234),(13)(24)} and S={id,(12)(34),(13)(24),(14)(23)} are not good. These results are strengthened by a very recent result of Kardoš and Narboni, which implies that S={id,(12)(34)} is not good and another very recent result of Zhu which implies that S={id,(12)} is not good. In this paper we prove if S is a subset of S4 containing id, then S is good if and only if S={id}.