33,248 research outputs found
Optimal Hamilton covers and linear arboricity for random graphs
In his seminal 1976 paper, P\'osa showed that for all , the
binomial random graph is with high probability Hamiltonian. This leads
to the following natural questions, which have been extensively studied: How
well is it typically possible to cover all edges of with Hamilton
cycles? How many cycles are necessary? In this paper we show that for , we can cover with precisely
Hamilton cycles. Our result is clearly best possible
both in terms of the number of required cycles, and the asymptotics of the edge
probability , since it starts working at the weak threshold needed for
Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and
improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of
Ferber, Kronenberg and Long, essentially closing a long line of research on
Hamiltonian packing and covering problems in random graphs.Comment: 13 page
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
On covering expander graphs by Hamilton cycles
The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree satisfies some basic expansion properties and contains
a family of edge disjoint Hamilton cycles, then there also
exists a covering of its edges by Hamilton cycles. This
implies that for every and every there exists
a covering of all edges of by Hamilton cycles
asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275
Hamilton cycles in quasirandom hypergraphs
We show that, for a natural notion of quasirandomness in -uniform
hypergraphs, any quasirandom -uniform hypergraph on vertices with
constant edge density and minimum vertex degree contains a
loose Hamilton cycle. We also give a construction to show that a -uniform
hypergraph satisfying these conditions need not contain a Hamilton -cycle
if divides . The remaining values of form an interesting
open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm
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