1,842 research outputs found
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
Factors and Connected Factors in Tough Graphs with High Isolated Toughness
In this paper, we show that every -tough graph with order and isolated
toughness at least has a factor whose degrees are , except for at most
one vertex with degree . Using this result, we conclude that every
-tough graph with order and isolated toughness at least has a
connected factor whose degrees lie in the set , where .
Also, we show that this factor can be found -tree-connected, when is a
-tough graph with order and isolated toughness at least ,
where and . Next, we prove that
every -tough graph of order at least with high enough
isolated toughness admits an -tree-connected factor with maximum degree at
most . From this result, we derive that every -tough graph
of order at least three with high enough isolated toughness has a spanning
Eulerian subgraph whose degrees lie in the set . In addition, we
provide a family of -tough graphs with high enough isolated toughness
having no connected even factors with bounded maximum degree
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