Facial packing vertex-coloring of subdivided plane graphs

Abstract

A facial packing vertex-coloring of a plane graph G is a coloring of its vertices with colors 1,…,k such that every facial path containing two vertices with the same color i has at least i+2 vertices. The smallest positive integer k such that G admits a facial packing vertex-coloring with colors 1,…,k is denoted by pf(G). Let Si(G) denote the graph obtained from G by subdividing each of its edges precisely i times, i≥0. In this paper we deal with a question whether pf(Si(G)) is bounded.