TY - JOUR
T1 - On γ-Vectors Satisfying the Kruskal-Katona Inequalities
AU - Nevo, Eran
AU - Petersen, Kyle K.
PY - 2011/4
Y1 - 2011/4
N2 - We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal-Katona inequalities but also the stronger Frankl-Füredi-Kalai inequalities. In another direction, we show that if a flag (d-1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl-Füredi-Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal-Katona, and further, the Frankl-Füredi-Kalai inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such γ-vectors are nonnegative.
AB - We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal-Katona inequalities but also the stronger Frankl-Füredi-Kalai inequalities. In another direction, we show that if a flag (d-1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl-Füredi-Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal-Katona, and further, the Frankl-Füredi-Kalai inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such γ-vectors are nonnegative.
KW - Gal's conjecture
KW - γ-Vector
UR - http://www.scopus.com/inward/record.url?scp=79751532326&partnerID=8YFLogxK
U2 - 10.1007/s00454-010-9243-6
DO - 10.1007/s00454-010-9243-6
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AN - SCOPUS:79751532326
SN - 0179-5376
VL - 45
SP - 503
EP - 521
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
IS - 3
ER -