Precoloring 3-extension on outerplanar graphs

TL;DR

This paper investigates the precoloring extension problem in outerplanar graphs with limited triangles, demonstrating that certain precolorings can always be extended to a 3-coloring.

Contribution

It extends previous results by showing that precolorings of specific non-adjacent vertices in outerplanar graphs with few triangles are extendable to a 3-coloring.

Findings

Precoloring of any two non-adjacent vertices can be extended.

Precoloring of any three non-adjacent vertices can be extended.

Results apply to outerplanar graphs with at most two triangles.

Abstract

The precoloring problem of a graph involves assigning colors to some vertices beforehand, and the objective is to determine whether it can be extended to a proper k-coloring of the entire graph. In 1958, Grotzsch proved that every triangle-free planar graph can be properly colored by three colors. One of the further generalizations of it is the recent result by Hoang La et al. in (Discrete Mathematics, 345(6) (2022), 112849 ). They proved that any two non-adjacent vertices and a face with a length at most four are precolored, the precolorings can be extended to a 3-coloring of the graph. In the paper, we consider precoloring extension of connected outerplanar graph with at most one or two triangles. Particularly, we show that precoloring of any two or three non-adjacent vertices can be extend to a 3-coloring of the whole graph.