655 research outputs found
Finding Even Cycles Faster via Capped k-Walks
In this paper, we consider the problem of finding a cycle of length (a
) in an undirected graph with nodes and edges for constant
. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that
if , then contains a , further implying that
one needs to consider only graphs with .
Previously the best known algorithms were an algorithm due to Yuster
and Zwick [J.Disc.Math'97] as well as a algorithm by Alon et al. [Algorithmica'97].
We present an algorithm that uses time and finds a
if one exists. This bound is exactly when . For
-cycles our new bound coincides with Alon et al., while for every our
bound yields a polynomial improvement in .
Yuster and Zwick noted that it is "plausible to conjecture that is
the best possible bound in terms of ". We show "conditional optimality": if
this hypothesis holds then our algorithm is tight as well.
Furthermore, a folklore reduction implies that no combinatorial algorithm can
determine if a graph contains a -cycle in time for any
under the widely believed combinatorial BMM conjecture. Coupled
with our main result, this gives tight bounds for finding -cycles
combinatorially and also separates the complexity of finding - and
-cycles giving evidence that the exponent of in the running time should
indeed increase with .
The key ingredient in our algorithm is a new notion of capped -walks,
which are walks of length that visit only nodes according to a fixed
ordering. Our main technical contribution is an involved analysis proving
several properties of such walks which may be of independent interest.Comment: To appear at STOC'1
Asymptotic multipartite version of the Alon-Yuster theorem
In this paper, we prove the asymptotic multipartite version of the
Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi
theorem: If is an integer, is a -colorable graph and
is fixed, then, for every sufficiently large , where
divides , and for every balanced -partite graph on vertices with
each of its corresponding bipartite subgraphs having minimum
degree at least , has a subgraph consisting of
vertex-disjoint copies of .
The proof uses the Regularity method together with linear programming.Comment: 22 pages, 1 figur
On embedding well-separable graphs
Call a simple graph of order well-separable, if by deleting a
separator set of size the leftover will have components of size at most
. We prove, that bounded degree well-separable spanning subgraphs are
easy to embed: for every and positive integer there exists
an such that if , for a well-separable
graph of order and
for a simple graph of order , then . We extend our result
to graphs with small band-width, too.Comment: 11 pages, submitted for publicatio
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
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