  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.
