An improved bound on parity vertex colourings of outerplane graphs

A parity vertex colouring of a 2-connected plane graph GG is a proper vertex colouring such that for each face ff and colour ii, either zero or an odd number of vertices incident with ff are coloured ii. The parity chromatic number χp(G)χp(G) of GG is the smallest number of colours used in a parity vertex colouring of GG.In this paper, we improve a result of Czap by showing that every 2-connected outerplane graph GG, with two exceptions, has χp(G)≤9χp(G)≤9. In addition, we characterize the 2-connected outerplane graphs GG with χp(G)=2χp(G)=2 and those which are bipartite and have χp(G)=8χp(G)=8.