Extending Colorings of Planar Graphs

Abstract

Let G be a planar graph and W be a subgraph whose each component is a $K_{2}$ of G. Let d be the minimum distance between any two distinct components of W. It is known that, if $d\ge 8$, then any 5-coloring of W can be extended to a 5-coloring of G. Up to now, it is the best possible with respect to the distance constraint. In this paper, we obtain sufficient conditions for such coloring extension when $d\ge 4$, $d\ge 6$ and $d\ge 7$. It partially answers Albertson’s question in Albertson (1998). Then we extend a certain 3-coloring of some disjoint cycles that are face boundaries to a 5-coloring of the entire graph.