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 inverted right perpendiculart-1/2inverted left perpendicular-2} many colors. If the input graph is triangle-free, we only need max {4, inverted right perpendiculart-1inverted right perpendicular/2+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.