550 research outputs found
Low-distortion embeddings of graphs with large girth
The main purpose of the paper is to construct a sequence of graphs of
constant degree with indefinitely growing girths admitting embeddings into
with uniformly bounded distortions. This result answers the problem
posed by N. Linial, A. Magen, and A. Naor (2002).Comment: Some confusing omissions are corrected in the second versio
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree
The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree there exists a natural number such that the following
holds: If is a -edge-connected simple graph with size divisible by
the size of , then can be edge-decomposed into subgraphs isomorphic to
. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from by vertex-identifications.
We call such a subgraph a homomorphic copy of . This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of
is greater than the diameter of . As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page
Design of Finite-Length Irregular Protograph Codes with Low Error Floors over the Binary-Input AWGN Channel Using Cyclic Liftings
We propose a technique to design finite-length irregular low-density
parity-check (LDPC) codes over the binary-input additive white Gaussian noise
(AWGN) channel with good performance in both the waterfall and the error floor
region. The design process starts from a protograph which embodies a desirable
degree distribution. This protograph is then lifted cyclically to a certain
block length of interest. The lift is designed carefully to satisfy a certain
approximate cycle extrinsic message degree (ACE) spectrum. The target ACE
spectrum is one with extremal properties, implying a good error floor
performance for the designed code. The proposed construction results in
quasi-cyclic codes which are attractive in practice due to simple encoder and
decoder implementation. Simulation results are provided to demonstrate the
effectiveness of the proposed construction in comparison with similar existing
constructions.Comment: Submitted to IEEE Trans. Communication
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