11,780 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
Hamilton cycles, minimum degree and bipartite holes
We present a tight extremal threshold for the existence of Hamilton cycles in
graphs with large minimum degree and without a large ``bipartite hole`` (two
disjoint sets of vertices with no edges between them). This result extends
Dirac's classical theorem, and is related to a theorem of Chv\'atal and
Erd\H{o}s.
In detail, an -bipartite-hole in a graph consists of two disjoint
sets of vertices and with and such that there are no
edges between and ; and is the maximum integer
such that contains an -bipartite-hole for every pair of
non-negative integers and with . Our central theorem is that
a graph with at least vertices is Hamiltonian if its minimum degree is
at least .
From the proof we obtain a polynomial time algorithm that either finds a
Hamilton cycle or a large bipartite hole. The theorem also yields a condition
for the existence of edge-disjoint Hamilton cycles. We see that for dense
random graphs , the probability of failing to contain many
edge-disjoint Hamilton cycles is . Finally, we discuss
the complexity of calculating and approximating
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
A remark on zeta functions of finite graphs via quantum walks
From the viewpoint of quantum walks, the Ihara zeta function of a finite
graph can be said to be closely related to its evolution matrix. In this note
we introduce another kind of zeta function of a graph, which is closely related
to, as to say, the square of the evolution matrix of a quantum walk. Then we
give to such a function two types of determinant expressions and derive from it
some geometric properties of a finite graph. As an application, we illustrate
the distribution of poles of this function comparing with those of the usual
Ihara zeta function.Comment: 14 pages, 1 figur
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