Colouring of generalized signed triangle-free planar graphs

Abstract

We view an undirected graph G as a symmetric digraph, where each edge xy is replaced by two opposite arcs e=(x,y) and e−1=(y,x). Assume S is an inverse closed subset of permutations of positive integers. We say G is S-k-colourable if for any mapping σ:E(G)→S with σ(x,y)=(σ(y,x))−1, there is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). The concept of S-k-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets S such that every triangle-free planar graph is S-3-colourable. Such a set S is called TFP-good. Grötzsch’s theorem is equivalent to say that S={id} is TFP-good. We prove that for any inverse closed subset S of S3 which is not isomorphic to {id,(12)}, S is TFP-good if and only if either S={id} or there exists a∈[3] such that for each π∈S, π(a)≠a. It remains an open question to determine whether or not S={id,(12)} is TFP-good.