265 research outputs found
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
Robust Factorizations and Colorings of Tensor Graphs
Since the seminal result of Karger, Motwani, and Sudan, algorithms for
approximate 3-coloring have primarily centered around SDP-based rounding.
However, it is likely that important combinatorial or algebraic insights are
needed in order to break the threshold. One way to develop new
understanding in graph coloring is to study special subclasses of graphs. For
instance, Blum studied the 3-coloring of random graphs, and Arora and Ge
studied the 3-coloring of graphs with low threshold-rank.
In this work, we study graphs which arise from a tensor product, which appear
to be novel instances of the 3-coloring problem. We consider graphs of the form
with and ,
where is any edge set such that no vertex has
more than an fraction of its edges in . We show that one can
construct with that is close to . For arbitrary , satisfies . Additionally when is a
mild expander, we provide a 3-coloring for in polynomial time. These
results partially generalize an exact tensor factorization algorithm of Imrich.
On the other hand, without any assumptions on , we show that it is NP-hard
to 3-color .Comment: 34 pages, 3 figure
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