Parity vertex coloring of outerplane graphs

A proper vertex coloring of a 2-connected plane graph G is a parity vertex coloring if for each face f and each color c , the total number of vertices of color c incident with f is odd or zero. The minimum number of colors used in such a coloring of G is denoted by χ p ( G ) . s paper we prove that χ p ( G ) ≤ 12 for every 2-connected outerplane graph G . We show that there is a 2-connected outerplane graph G such that χ p ( G ) = 10 . If a 2-connected outerplane graph G is bipartite, then χ p ( G ) ≤ 8 , moreover, this bound is best possible.