Approximately coloring graphs without long induced paths

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 $O((3^{{t-2}+t2)m+n)$} if the input graph has $n$ vertices and $m$ edges.