Approximately coloring graphs without long induced paths

TL;DR

This paper presents an algorithm for coloring graphs without long induced paths, providing bounds on the number of colors needed and analyzing its efficiency, advancing understanding of graph coloring complexity in restricted classes.

Contribution

It introduces a polynomial-time algorithm for coloring graphs without long induced paths using a bounded number of colors, depending on the path length and triangle-free property.

Findings

Algorithm computes colorings with bounded colors based on path length

Color bounds are tighter for triangle-free graphs

Running time is polynomial in graph size and path length

Abstract

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.