84,681 research outputs found
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
The critical window for the classical Ramsey-Tur\'an problem
The first application of Szemer\'edi's powerful regularity method was the
following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any
K_4-free graph on N vertices with independence number o(N) has at most (1/8 +
o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising
geometric construction, utilizing the isoperimetric inequality for the high
dimensional sphere, of a K_4-free graph on N vertices with independence number
o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in
1976, several problems have been asked on estimating the minimum possible
independence number in the critical window, when the number of edges is about
N^2 / 8. These problems have received considerable attention and remained one
of the main open problems in this area. In this paper, we give nearly
best-possible bounds, solving the various open problems concerning this
critical window.Comment: 34 page
A Quantum Lovasz Local Lemma
The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show
the existence of combinatorial objects meeting a prescribed collection of
"weakly dependent" criteria. We show that the LLL extends to a much more
general geometric setting, where events are replaced with subspaces and
probability is replaced with relative dimension, which allows to lower bound
the dimension of the intersection of vector spaces under certain independence
conditions. Our result immediately applies to the k-QSAT problem: For instance
we show that any collection of rank 1 projectors with the property that each
qubit appears in at most of them, has a joint satisfiable
state.
We then apply our results to the recently studied model of random k-QSAT.
Recent works have shown that the satisfiable region extends up to a density of
1 in the large k limit, where the density is the ratio of projectors to qubits.
Using a hybrid approach building on work by Laumann et al. we greatly extend
the known satisfiable region for random k-QSAT to a density of
. Since our tool allows us to show the existence of joint
satisfying states without the need to construct them, we are able to penetrate
into regions where the satisfying states are conjectured to be entangled,
avoiding the need to construct them, which has limited previous approaches to
product states.Comment: 19 page
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