List-three-coloring graphs with no induced $P_6+rP_3$

TL;DR

This paper presents a polynomial-time algorithm for testing 3-colorability and list 3-colorability of graphs that do not contain an induced subgraph isomorphic to $P_6+rP_3$, expanding understanding of coloring in restricted graph classes.

Contribution

It introduces the first polynomial-time algorithms for 3-coloring and list 3-coloring in graphs excluding $P_6+rP_3$ as an induced subgraph.

Findings

Polynomial-time algorithm for 3-colorability

Algorithm for list 3-colorability

Applicable to graphs with no induced $P_6+rP_3$

Abstract

For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In this paper we provide a polynomial-time algorithm to test if a graph with no induced subgraph isomorphic to $P_6+rP_3$ is three-colorable. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$.