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It is an open problem whether the 3-coloring problem can be solved inpolynomial 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-colorablegraph 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 istriangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ manycolors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the inputgraph has $n$ vertices and $m$ edges.

Approximately coloring graphs without long induced paths