27 research outputs found
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton
cycle, which is applicable to a wide variety of graphs, including relatively
sparse graphs. In contrast to previous criteria, ours is based on only two
properties: one requiring expansion of ``small'' sets, the other ensuring the
existence of an edge between any two disjoint ``large'' sets. We also discuss
applications in positional games, random graphs and extremal graph theory.Comment: 19 page
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on two properties only: one requiring expansion of "smallâ sets, the other ensuring the existence of an edge between any two disjoint "largeâ sets. We also discuss applications in positional games, random graphs and extremal graph theor
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
Optimal covers with Hamilton cycles in random graphs
A packing of a graph G with Hamilton cycles is a set of edge-disjoint
Hamilton cycles in G. Such packings have been studied intensively and recent
results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has
size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o recently
initiated research on the `dual' problem, where one asks for a set of Hamilton
cycles covering all edges of G. Our main result states that for log^{117}n / n
< p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil
Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an
approximate result of Glebov, Krivelevich and Szab\'o, which holds for p >
n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on
packing Hamilton cycles in pseudorandom graphs.Comment: final version of paper (to appear in Combinatorica
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
Sharp threshold for embedding combs and other spanning trees in random graphs
When , the tree consists of a path containing
vertices, each of whose vertices has a disjoint path length
beginning at it. We show that, for any and , the binomial
random graph almost surely contains
as a subgraph. This improves a recent result of Kahn,
Lubetzky and Wormald. We prove a similar statement for a more general class of
trees containing both these combs and all bounded degree spanning trees which
have at least disjoint bare paths length .
We also give an efficient method for finding large expander subgraphs in a
binomial random graph. This allows us to improve a result on almost spanning
trees by Balogh, Csaba, Pei and Samotij.Comment: 20 page