3,265 research outputs found
Sprinkling with random regular graphs
We conjecture that the distribution of the edge-disjoint union of two random
regular graphs on the same vertex set is asymptotically equivalent to a random
regular graph of the combined degree, provided that the combined degree and the
complementary degrees are growing. We verify this conjecture for the cases when
the graphs are sufficiently dense or sparse. We also prove an asymptotic
formula for the expected number of spanning regular subgraphs in a random
regular graph
Hamilton decompositions of regular tournaments
We show that every sufficiently large regular tournament can almost
completely be decomposed into edge-disjoint Hamilton cycles. More precisely,
for each \eta>0 every regular tournament G of sufficiently large order n
contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an
approximate solution to a conjecture of Kelly from 1968. Our result also
extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our
main result. To appear in the Proceedings of the LM
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
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of
well known classical results in extremal graph theory. Here we follow this
trend and investigate the structure of triangle-free subgraphs of with
high minimum degree. We prove that asymptotically almost surely each
triangle-free spanning subgraph of with minimum degree at least
is -close to bipartite,
and each spanning triangle-free subgraph of with minimum degree at
least is -close to
-partite for some . These are random graph analogues of a
result by Andr\'asfai, Erd\H{o}s, and S\'os [Discrete Math. 8 (1974), 205-218],
and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show
that our results are best possible up to a constant factor.Comment: 18 page
Enumeration of spanning trees in a pseudofractal scale-free web
Spanning trees are an important quantity characterizing the reliability of a
network, however, explicitly determining the number of spanning trees in
networks is a theoretical challenge. In this paper, we study the number of
spanning trees in a small-world scale-free network and obtain the exact
expressions. We find that the entropy of spanning trees in the studied network
is less than 1, which is in sharp contrast to previous result for the regular
lattice with the same average degree, the entropy of which is higher than 1.
Thus, the number of spanning trees in the scale-free network is much less than
that of the corresponding regular lattice. We present that this difference lies
in disparate structure of the two networks. Since scale-free networks are more
robust than regular networks under random attack, our result can lead to the
counterintuitive conclusion that a network with more spanning trees may be
relatively unreliable.Comment: Definitive version accepted for publication in EPL (Europhysics
Letters
On the threshold for k-regular subgraphs of random graphs
The -core of a graph is the largest subgraph of minimum degree at least
. We show that for sufficiently large, the -core of a random
graph \G(n,p) asymptotically almost surely has a spanning -regular
subgraph. Thus the threshold for the appearance of a -regular subgraph of a
random graph is at most the threshold for the -core. In particular, this
pins down the point of appearance of a -regular subgraph in \G(n,p) to a
window for of width roughly for large and moderately large
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