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 : V-G -> (1,...,k) such that c(u) not equal 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 l(g), such that every graph with girth at least g and with no induced path on l(g) vertices is 3-colorable. We also show that for all fixed integers k, >= 1, the k-COLORING 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, l >= 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 l vertices. This result is best possible, as we prove the existence of an integer l*, such that already 4-COLORING is NP-complete for graphs with girth 4 and with no induced path on,l* vertices. (C) 2013 Elsevier B.V. All rights reserved.