6,953 research outputs found
Product Dimension of Forests and Bounded Treewidth Graphs
The product dimension of a graph G is defined as the minimum natural number l
such that G is an induced subgraph of a direct product of l complete graphs. In
this paper we study the product dimension of forests, bounded treewidth graphs
and k-degenerate graphs. We show that every forest on n vertices has a product
dimension at most 1.441logn+3. This improves the best known upper bound of
3logn for the same due to Poljak and Pultr. The technique used in arriving at
the above bound is extended and combined with a result on existence of
orthogonal Latin squares to show that every graph on n vertices with a
treewidth at most t has a product dimension at most (t+2)(logn+1). We also show
that every k-degenerate graph on n vertices has a product dimension at most
\ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by
Eaton and Rodl.Comment: 12 pages, 3 figure
Wide partitions, Latin tableaux, and Rota's basis conjecture
Say that mu is a ``subpartition'' of an integer partition lambda if the
multiset of parts of mu is a submultiset of the parts of lambda, and define an
integer partition lambda to be ``wide'' if for every subpartition mu of lambda,
mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of
mu). Then Brian Taylor and the first author have conjectured that an integer
partition lambda is wide if and only if there exists a tableau of shape lambda
such that (1) for all i, the entries in the ith row of the tableau are
precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the
entries in the jth column of the tableau are pairwise distinct. This conjecture
was originally motivated by Rota's basis conjecture and, if true, yields a new
class of integer multiflow problems that satisfy max-flow min-cut and
integrality. Wide partitions also yield a class of graphs that satisfy
``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above
conjecture implies that these graphs furthermore have a completely saturated
stable set partition. We present several partial results, but the conjecture
remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results;
28 pages, submitted to Adv. Appl. Mat
The Bounded Edge Coloring Problem and Offline Crossbar Scheduling
This paper introduces a variant of the classical edge coloring problem in
graphs that can be applied to an offline scheduling problem for crossbar
switches. We show that the problem is NP-complete, develop three lower bounds
bounds on the optimal solution value and evaluate the performance of several
approximation algorithms, both analytically and experimentally. We show how to
approximate an optimal solution with a worst-case performance ratio of
and our experimental results demonstrate that the best algorithms produce
results that very closely track a lower bound
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
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