Coloring graphs without short cycles and long induced paths.

Abstract

The girth of a graph G is the length of a shortest cycle in G. For any fixed girth g ≥ 4 we determine a lower bound ℓ(g) such that every graph with girth at least g and with no induced path on ℓ(g) vertices is 3-colorable. In contrast, we show the existence of an integer ℓ such that testing for 4-colorability is NP-complete for graphs with girth 4 and with no induced path on ℓ vertices.