Colorful induced subgraphs

Kierstead, H.A. and W.T. Trotter. Colorful induced subgraphs, Discrete Mathematics 101 (1992) 165-169. A colored graph is a graph whose vertices have been properly, though not necessarily optimally colored, with integers. Colored graphs have a natural orientation in which edges are directed from the end point with smaller color to the end point with larger color. A subgraph of a colored graph is colorful if each of its vertices has a distinct color. We prove that there exists a function f (k, n) such that for any colored graph G, if x(G) > f (w(G), n) then G induces either a colorful out directed star with n leaves or a colorful directed path on n vertices. We also show that this result would be false if either alternative was omitted. Our results provide a solution to Problem 115. Discrete Math. 79.