Coloring Graphs without Short Cycles and Long Induced Paths

For an integer k ≥ 1, a graph G is k-colorable if there exists a mapping c : VG → {1, . . . , k} such that c(u) 6= c(v) whenever u and v are two adjacent vertices. For a fixed integer k ≥ 1, the k-COLORING problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g ≥ 4 we determine a lower bound `(g), such that every graph with girth at least g and with no induced path on `(g) vertices is 3-colorable. We also show that for all fixed integers k, ` ≥ 1, the kCOLORING problem can be solved in polynomial time for graphs with no induced cycle on four vertices and no induced path on ` vertices. As a consequence, for all fixed integers k, ` ≥ 1 and g ≥ 5, the k-COLORING problem can be solved in polynomial time for graphs with girth at least g and with no induced path on ` vertices. This result is best possible, as we prove the existence of an integer `∗, such that already 4-COLORING is NP-complete for graphs with girth 4 and with no induced path on `∗ vertices.