36,844 research outputs found
2-factors with k cycles in Hamiltonian graphs
A well known generalisation of Dirac's theorem states that if a graph on
vertices has minimum degree at least then contains a
-factor consisting of exactly cycles. This is easily seen to be tight in
terms of the bound on the minimum degree. However, if one assumes in addition
that is Hamiltonian it has been conjectured that the bound on the minimum
degree may be relaxed. This was indeed shown to be true by S\'ark\"ozy. In
subsequent papers, the minimum degree bound has been improved, most recently to
by DeBiasio, Ferrara, and Morris. On the other hand no
lower bounds close to this are known, and all papers on this topic ask whether
the minimum degree needs to be linear. We answer this question, by showing that
the required minimum degree for large Hamiltonian graphs to have a -factor
consisting of a fixed number of cycles is sublinear in Comment: 13 pages, 6 picture
Optimal General Matchings
Given a graph and for each vertex a subset of the
set , where denotes the degree of vertex
in the graph , a -factor of is any set such that
for each vertex , where denotes the number of
edges of incident to . The general factor problem asks the existence of
a -factor in a given graph. A set is said to have a {\em gap of
length} if there exists a natural number such that and . Without any restrictions the
general factor problem is NP-complete. However, if no set contains a gap
of length greater than , then the problem can be solved in polynomial time
and Cornuejols \cite{Cor} presented an algorithm for finding a -factor, if
it exists. In this paper we consider a weighted version of the general factor
problem, in which each edge has a nonnegative weight and we are interested in
finding a -factor of maximum (or minimum) weight. In particular, this
version comprises the minimum/maximum cardinality variant of the general factor
problem, where we want to find a -factor having a minimum/maximum number of
edges.
We present an algorithm for the maximum/minimum weight -factor for the
case when no set contains a gap of length greater than . This also
yields the first polynomial time algorithm for the maximum/minimum cardinality
-factor for this case
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
Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts
We seek to augment a geometric network in the Euclidean plane with shortcuts
to minimize its continuous diameter, i.e., the largest network distance between
any two points on the augmented network. Unlike in the discrete setting where a
shortcut connects two vertices and the diameter is measured between vertices,
we take all points along the edges of the network into account when placing a
shortcut and when measuring distances in the augmented network.
We study this network augmentation problem for paths and cycles. For paths,
we determine an optimal shortcut in linear time. For cycles, we show that a
single shortcut never decreases the continuous diameter and that two shortcuts
always suffice to reduce the continuous diameter. Furthermore, we characterize
optimal pairs of shortcuts for convex and non-convex cycles. Finally, we
develop a linear time algorithm that produces an optimal pair of shortcuts for
convex cycles. Apart from the algorithms, our results extend to rectifiable
curves.
Our work reveals some of the underlying challenges that must be overcome when
addressing the discrete version of this network augmentation problem, where we
minimize the discrete diameter of a network with shortcuts that connect only
vertices
Long properly colored cycles in edge colored complete graphs
Let denote a complete graph on vertices whose edges are
colored in an arbitrary way. Let denote the
maximum number of edges of the same color incident with a vertex of
. A properly colored cycle (path) in is a cycle (path)
in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s
(1976) proposed the following conjecture: if , then contains a properly
colored Hamiltonian cycle. Li, Wang and Zhou proved that if
, then
contains a properly colored cycle of length at least . In this paper, we improve the bound to .Comment: 8 page
- …