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 $$\max \left\{ 5,2\left\lceil \frac{t-1}{2}\right\rceil -2\right\} $$max5,2t-12-2 many colors. If the input graph is triangle-free, we only need $$\max \left\{ 4,\left\lceil \frac{t-1}{2}\right\rceil +1\right\} $$max4,t-12+1 many colors. The running time of our algorithm is $$O((3^{t-2}+t^2)m+n)$$O((3t-2+t2)m+n) if the input graph has n vertices and m edges.