Colouring of generalized signed triangle-free planar graphs

Abstract We view an undirected graph G as a symmetric digraph, where each edge x y 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 σ e ( 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. Grotzsch’s theorem is equivalent to say that S = { i d } is TFP-good. We prove that for any inverse closed subset S of S 3 which is not isomorphic to { i d , ( 12 ) } , S is TFP-good if and only if either S = { i d } 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 = { i d , ( 12 ) } is TFP-good.