### Path 3-coloring plane graphs

Jun 15, 2018

A path coloring of a graph partitions its vertex set into color classes such
that each class induces a disjoint union of paths. In my computer science masters
project I implemented
several algorithms to compute path colorings of graphs embedded in the plane.

### Abstract

We present two algorithms to path color plane graphs with 3 colors
based on a proof by Poh in 1990. First we consider a naive algorithm that
directly follows Poh’s procedure, then we modify the algorithm
to run in linear time.

Independent results of Hartman and Skrekovski describe a procedure that takes
a plane graph $G$ and a list of 3 colors for each vertex, and
computes a path coloring of $G$ such that each vertex receives a color from its
list. We implement a linear time algorithm based on Hartman and Skrekovski’s
proofs.

A C++ implementation is provided for all three algorithms, utilizing the Boost
Graph Library. Instructions are given on how to use the implementation
to construct colorings for plane graphs represented by Boost data
types.

### Links