3,756 research outputs found
Almost-spanning universality in random graphs (Extended abstract)
A graph G is said to be ℋ(n, Δ)-universal if it contains every graph on n vertices with maximum degree at most Δ. It is known that for any ε > 0 and any natural number Δ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely ℋ((1 - ε)n, Δ)-universal for p ≥ c(log n/n)^(1/Δ). Bypassing this natural boundary Δ ≥ 3, we show that for the same conclusion holds when [equation; see abstract in PDF for details]
Almost spanning subgraphs of random graphs after adversarial edge removal
Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with
p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost
spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth
in the following sense: asymptotically almost surely, if an adversary deletes
arbitrary edges in G(n,p) such that each vertex loses less than half of its
neighbours, then the resulting graph still contains a copy of all such H.Comment: 46 pages, 6 figure
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Powers of Hamilton cycles in pseudorandom graphs
We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . This improves on a result of
Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random
graphs, Combinatorica 24 (2004), no. 3, 403--426].
We also extend our result to higher powers of Hamilton cycles and establish
corresponding counting versions.Comment: 30 pages, 1 figur
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