Complements of nearly perfect graphs

A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A. Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$, we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $\chi$-bounded

Property (T)(T) and strong Property (T)(T) for unital C∗C^*-algebras

In this paper, we will give a thorough study of the notion of Property $(T)$ for $C^*$-algebras (as introduced by M.B. Bekka in \cite{Bek-T}) as well as a slight stronger version of it, called "strong property $(T)$" (which is also an analogue of the corresponding concept in the case of discrete groups and type $\rm II_1$-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property $(T)$ and strong property $(T)$. We will also relate them to certain $(T)$-type properties of the unitary group of the underlying $C^*$-algebra

SO(2,C) Invariant Discrete Gauge States in Liouville Gravity Coupled to Minimal Conformal Matter

We contruct the general formula for a set of discrete gauge states (DGS) in c<1 Liouville theory. This formula reproduces the previously found c=1 DGS in the appropriate limiting case. We also demonstrate the SO(2,C) invariant structure of these DGS in the old covariant quantization of the theory. This is in analogy to the SO(2,C) invariant ring structure of BRST cohomology of the theory

Precise Predictions for Higgs Production in Neutralino Decays

Complete one-loop results, supplemented by two-loop Higgs propagator-type corrections, are obtained for the class of processes chi^0_i->chi^0_j h_a in the MSSM with CP-violating phases for parameters entering the process beyond lowest order. The parameter region of the CPX benchmark scenario where a very light Higgs boson is unexcluded by present data is analysed in detail. We find that the decay chi^0_2->chi^0_1 h_1 may offer good prospects to detect such a light Higgs boson.Comment: 4 pages, 3 figures; to appear in the proceedings of the 17th International Conference on Supersymmetry and the Unification of Fundamental Interactions (SUSY09), Boston, USA, 5-10 Jun 200