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 $\max \{5,2\lceil \frac{t-1}{2}\rceil -2\}$ 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.