An improved bound on parity vertex colourings of outerplane graphs

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