" " S) ~{ On Greene's Theorem for Digraphs - Irith Ben-Arroyo Hartman Fathi Saleh Daniel Hershkowitz DEPARTMENT OF MA THEMATICS TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL ABSTRACT Greene's Theorem states that the maximum cardinality of an optimal k-path in a poset is equal to the minimum k-norm of a k-optimal coloring. This result was extended to all acyclic digraphs, and is conjectured to hold for general digraphs. We prove the result for general digraphs in which an optimal k-path contains a path of cardinality one. This implies the validity of the conjecture for all bipartite digraphs. We also extend Greene's Theorem to all split graphs. @ 1994 John Wiley & Sons, Inc. 1. INTRODUCTION Let D = (V, E) be a digraph. A path in D is a sequence of distinct vertices (VI,V2,...,Vh) such that (Vj,Vj+l) E E, i = 1,2,...,h - 1. The set of vertices {VI,V2,...,Vh} of the path P =.(VI,V2,...,Vh) is denoted by V(P). The cardinality of P, denoted by IPI, is IV(P)I. A k-path pk = {PI,P2,...,Pt} is a family of at most k vertex disjoint paths Pi, The cardinality IPkl of a k-path pk = {PI, P2,. .., PI} is IU~=1 V(Pj)l. We say that pk is optimal if IU~=1 V(Pj)1 is as large as possible. An independent set is a set of vertices, no two of which are adjacent. A coloring C is a partition of V into disjoint independent sets. Each independent set in C is also called a color class. For each nonnegative integer k, the k-norm IClk of a coloring C = {CI, C2,..., Cm} is defined by IClk = ~~=l min{ICjl, k}. A coloring that minimizes IClk is called k-optimal. For example, a I-optimal coloring is a coloring with X colors, where X is the chromatic number of D. A coloring C = {CI, C2,..., Cm} and a k-path pk are orthogonal if every color class Cj in C meets min{ICjl, k} different paths of pt. Aharoni, Ben-Arroyo Hartman, and Hoffman made the following conjecture. Journal of Graph Theory, Vol. 18, No.2, 169-175 (1994) @ 1994 John Wiley & Sons, Inc. CCC 0364-9024/94/020169-07 i