Facial parity edge colouring of plane pseudographs

A facial parity edge colouring of a connected bridgeless plane graph is such an edge colouring in which no two face-adjacent edges receive the same colour and, in addition, for each face f and each colour c , either no edge or an odd number of edges incident with f is coloured with c . Let χ p ′ ( G ) denote the minimum number of colours used in such a colouring of G . In this paper we prove that χ p ′ ( G ) ≤ 20 for any 2-edge-connected plane graph G . In the case when G is a 3 -edge-connected plane graph the upper bound for this parameter is 12 . For G being 4 -edge-connected plane graph we have χ p ′ ( G ) ≤ 9 . On the other hand we prove that some bridgeless plane graphs require at least 10 colours for such a colouring.