524 research outputs found
Perfect packings with complete graphs minus an edge
Let K_r^- denote the graph obtained from K_r by deleting one edge. We show
that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that
every graph G whose order n\ge n_0 is divisible by r and whose minimum degree
is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a
collection of disjoint copies of K_r^- which covers all vertices of G. Here
chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The
bound on the minimum degree is best possible and confirms a conjecture of
Kawarabayashi for large n
Connectivity and Cycles
https://digitalcommons.memphis.edu/speccoll-faudreerj/1191/thumbnail.jp
Superconcentrators
An -superconcentrator is an acyclic directed graph with inputs and outputs for which, for every , every set of inputs, and every set of outputs, there exists an -flow (a set of vertex-disjoint directed paths) from the given inputs to the given outputs. We show that there exist -superconcentrators with (in fact, at most ) edges, depth , and maximum degree (in-degree plus out-degree) 16
Rigid partitions: from high connectivity to random graphs
A graph is called -rigid if there exists a generic embedding of its vertex
set into such that every continuous motion of the vertices that
preserves the lengths of all edges actually preserves the distances between all
pairs of vertices. The rigidity of a graph is the maximal such that the
graph is -rigid. We present new sufficient conditions for the -rigidity
of a graph in terms of the existence of ``rigid partitions'' -- partitions of
the graph that satisfy certain connectivity properties. This extends previous
results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz.
As an application, we present new results on the rigidity of highly-connected
graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense
graphs. In particular, we prove that random -regular graphs are
typically -rigid, demonstrate the existence of a giant -rigid component
in sparse random binomial graphs, and show that the rigidity of relatively
sparse random binomial bipartite graphs is roughly the same as that of the
complete bipartite graph, which we consider an interesting phenomenon.
Furthermore, we show that a graph admitting disjoint connected
dominating sets is -rigid. This implies a weak version of the
Lov\'asz--Yemini conjecture on the rigidity of highly-connected graphs. We also
present an alternative short proof for a recent result by Lew, Nevo, Peled, and
Raz, which asserts that the hitting time for -rigidity in the random graph
process typically coincides with the hitting time for minimum degree .Comment: 30 pages. In this updated version, we have added a theorem concerning
the rigidity of dense graphs and incorporated references to Vill\'anyi's
recent resolution of the Lov\'asz-Yemini conjectur
Selection Networks
An upper bound asymptotic to is established for the number of comparators required in a network that classifies values into two classes, each containing values, with each value in one class less than or equal to each value in the other. (The best lower bound known for this problem is asymptotic to .
Finite-rate sparse quantum codes aplenty
We introduce a methodology for generating random multi-qubit stabilizer codes
based on solving a constraint satisfaction problem (CSP) on random bipartite
graphs. This framework allows us to enforce stabilizer commutation, X/Z
balancing, finite rate, sparsity, and maximum-degree constraints simultaneously
in a CSP that we can then solve numerically. Using a state-of-the-art CSP
solver, we obtain convincing evidence for the existence of a satisfiability
threshold. Furthermore, the extent of the satisfiable phase increases with the
number of qubits. In that phase, finding sparse codes becomes an easy problem.
Moreover, we observe that the sparse codes found in the satisfiable phase
practically achieve the channel capacity for erasure noise. Our results show
that intermediate-size finite-rate sparse quantum codes are easy to find, while
also demonstrating a flexible methodology for generating good codes with custom
properties. We therefore establish a complete and customizable pipeline for
random quantum code discovery that can be geared towards near to mid-term
quantum processor layouts
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