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 ⌈ t−1 2 ⌉ − 2 } many colors. If the input graph is triangle-free, we only need max { 4, ⌈ t−1 2 ⌉ + 1 } many colors. The running time of our algorithm is O((3t−2 + t2)m + n) if the input graph has n vertices and m edges.