Approximately Coloring Graphs Without Long Induced Paths

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⌈t−12⌉−2} many colors. If the input graph is triangle-free, we only need max{4,⌈t−12⌉+1} many colors. The running time of our algorithm is O((3t−2+t2)m+n) if the input graph has n vertices and m edges.