论文信息 - On Stanley's chromatic symmetric function and clawfree graphs

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.

Vesselin Gasharov

[1] Bruce E. Sagan,et al. Inductive proofs of q-log concavity , 1992, Discret. Math..

[2] Vesselin Gasharov,et al. Incomparability graphs of (3 + 1)-free posets are s-positive , 1996, Discret. Math..

[3] Richard P. Stanley,et al. Graph colorings and related symmetric functions: ideas and applications A description of results, interesting applications, & notable open problems , 1998, Discret. Math..

[4] Yahya Ould Hamidoune. On the numbers of independent k-sets in a claw free graph , 1990, J. Comb. Theory, Ser. B.

[5] Richard P. Stanley,et al. A Symmetric Function Generalization of the Chromatic Polynomial of a Graph , 1995 .

[6] R. Stanley. What Is Enumerative Combinatorics , 1986 .

[7] O. J. Heilmann,et al. Theory of monomer-dimer systems , 1972 .

[8] Christian Krattenthaler,et al. Combinatorial Proof of the Log-Concavity of the Sequence of Matching Numbers , 1996, J. Comb. Theory, Ser. A.