On Stanley's chromatic symmetric function and clawfree graphs

Abstract

Stanley associated to a simple graph G a symmetric function X G which generalizes the chromatic polynomial of G. He conjectured that X G is Schur positive when G is clawfree. This is equivalent to the minors of a certain matrix with polynomial entries being polynomials with nonnegative coefficients. We prove this for the 2×2 minors, which extends a result of Krattenthaler (J. Combin. Theory Ser. A 74(2) (1996) 351–354). We also give a characterization of clawfree graphs in terms of cardinalities of stable sets.