2,343 research outputs found
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 some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
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 sparse robustly expanding digraphs
The notion of robust expansion has played a central role in the solution of
several conjectures involving the packing of Hamilton cycles in graphs and
directed graphs. These and other results usually rely on the fact that every
robustly expanding (di)graph with suitably large minimum degree contains a
Hamilton cycle. Previous proofs of this require Szemer\'edi's Regularity Lemma
and so this fact can only be applied to dense, sufficiently large robust
expanders. We give a proof that does not use the Regularity Lemma and, indeed,
we can apply our result to suitable sparse robustly expanding digraphs.Comment: Accepted for publication in The Electronic Journal of Combinatoric
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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