74,341 research outputs found
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
Stable multivariate -Eulerian polynomials
We prove a multivariate strengthening of Brenti's result that every root of
the Eulerian polynomial of type is real. Our proof combines a refinement of
the descent statistic for signed permutations with the notion of real
stability-a generalization of real-rootedness to polynomials in multiple
variables. The key is that our refined multivariate Eulerian polynomials
satisfy a recurrence given by a stability-preserving linear operator. Our
results extend naturally to colored permutations, and we also give stable
generalizations of recent real-rootedness results due to Dilks, Petersen, and
Stembridge on affine Eulerian polynomials of types and . Finally,
although we are not able to settle Brenti's real-rootedness conjecture for
Eulerian polynomials of type , nor prove a companion conjecture of Dilks,
Petersen, and Stembridge for affine Eulerian polynomials of types and ,
we indicate some methods of attack and pose some related open problems.Comment: 17 pages. To appear in J. Combin. Theory Ser.
Parking Spaces
Let be a Weyl group with root lattice and Coxeter number . The
elements of the finite torus are called the -{\sf parking
functions}, and we call the permutation representation of on the set of
-parking functions the (standard) -{\sf parking space}. Parking spaces
have interesting connections to enumerative combinatorics, diagonal harmonics,
and rational Cherednik algebras. In this paper we define two new -parking
spaces, called the {\sf noncrossing parking space} and the {\sf algebraic
parking space}, with the following features: 1) They are defined more generally
for real reflection groups. 2) They carry not just -actions, but -actions, where is the cyclic subgroup of generated by a Coxeter
element. 3) In the crystallographic case, both are isomorphic to the standard
-parking space. Our Main Conjecture is that the two new parking spaces are
isomorphic to each other as permutation representations of . This
conjecture ties together several threads in the Catalan combinatorics of finite
reflection groups. We provide evidence for the conjecture, proofs of some
special cases, and suggest further directions for the theory.Comment: 49 pages, 10 figures, Version to appear in Advances in Mathematic
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