66,648 research outputs found
The (2k-1)-connected multigraphs with at most k-1 disjoint cycles
In 1963, Corr\'adi and Hajnal proved that for all and ,
every (simple) graph on n vertices with minimum degree at least 2k contains k
disjoint cycles. The same year, Dirac described the 3-connected multigraphs not
containing two disjoint cycles and asked the more general question: Which
(2k-1)-connected multigraphs do not contain k disjoint cycles? Recently, the
authors characterized the simple graphs G with minimum degree that do not contain k disjoint cycles. We use this result to answer
Dirac's question in full.Comment: 7 pages, 2 figures. To appear in Combinatoric
Hamilton decompositions of regular expanders: applications
In a recent paper, we showed that every sufficiently large regular digraph G
on n vertices whose degree is linear in n and which is a robust outexpander has
a decomposition into edge-disjoint Hamilton cycles. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.Comment: final version, to appear in J. Combinatorial Theory
Bose-Hubbard model on two-dimensional line graphs
We construct a basis for the many-particle ground states of the positive
hopping Bose-Hubbard model on line graphs of finite 2-connected planar
bipartite graphs at sufficiently low filling factors. The particles in these
states are localized on non-intersecting vertex-disjoint cycles of the line
graph which correspond to non-intersecting edge-disjoint cycles of the original
graph. The construction works up to a critical filling factor at which the
cycles are close-packed.Comment: 9 pages, 5 figures, figures and conclusions update
Edge-disjoint Hamilton cycles in graphs
In this paper we give an approximate answer to a question of Nash-Williams
from 1970: we show that for every \alpha > 0, every sufficiently large graph on
n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8
edge-disjoint Hamilton cycles. More generally, we give an asymptotically best
possible answer for the number of edge-disjoint Hamilton cycles that a graph G
with minimum degree \delta must have. We also prove an approximate version of
another long-standing conjecture of Nash-Williams: we show that for every
\alpha > 0, every (almost) regular and sufficiently large graph on n vertices
with minimum degree at least can be almost decomposed into
edge-disjoint Hamilton cycles.Comment: Minor Revisio
Color the cycles
The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013
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