10,703 research outputs found
Minimum Sum Edge Colorings of Multicycles
In the minimum sum edge coloring problem, we aim to assign natural numbers to
edges of a graph, so that adjacent edges receive different numbers, and the sum
of the numbers assigned to the edges is minimum. The {\em chromatic edge
strength} of a graph is the minimum number of colors required in a minimum sum
edge coloring of this graph. We study the case of multicycles, defined as
cycles with parallel edges, and give a closed-form expression for the chromatic
edge strength of a multicycle, thereby extending a theorem due to Berge. It is
shown that the minimum sum can be achieved with a number of colors equal to the
chromatic index. We also propose simple algorithms for finding a minimum sum
edge coloring of a multicycle. Finally, these results are generalized to a
large family of minimum cost coloring problems
Dependent Random Graphs and Multiparty Pointer Jumping
We initiate a study of a relaxed version of the standard Erdos-Renyi random
graph model, where each edge may depend on a few other edges. We call such
graphs "dependent random graphs". Our main result in this direction is a
thorough understanding of the clique number of dependent random graphs. We also
obtain bounds for the chromatic number. Surprisingly, many of the standard
properties of random graphs also hold in this relaxed setting. We show that
with high probability, a dependent random graph will contain a clique of size
, and the chromatic number will be at most
.
As an application and second main result, we give a new communication
protocol for the k-player Multiparty Pointer Jumping (MPJ_k) problem in the
number-on-the-forehead (NOF) model. Multiparty Pointer Jumping is one of the
canonical NOF communication problems, yet even for three players, its
communication complexity is not well understood. Our protocol for MPJ_3 costs
communication, improving on a bound of Brody
and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer
jumping problem , achieving an upper bound which is o(n) for
any players. This is the first o(n) bound for and
improves on a bound of Damm, Jukna, and Sgall [DJS98] which has stood for
almost twenty years.Comment: 18 page
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
Sum Coloring : New upper bounds for the chromatic strength
The Minimum Sum Coloring Problem (MSCP) is derived from the Graph Coloring
Problem (GCP) by associating a weight to each color. The aim of MSCP is to find
a coloring solution of a graph such that the sum of color weights is minimum.
MSCP has important applications in fields such as scheduling and VLSI design.
We propose in this paper new upper bounds of the chromatic strength, i.e. the
minimum number of colors in an optimal solution of MSCP, based on an
abstraction of all possible colorings of a graph called motif. Experimental
results on standard benchmarks show that our new bounds are significantly
tighter than the previous bounds in general, allowing to reduce substantially
the search space when solving MSCP .Comment: pre-prin
Boolean complexes for Ferrers graphs
In this paper we provide an explicit formula for calculating the boolean
number of a Ferrers graph. By previous work of the last two authors, this
determines the homotopy type of the boolean complex of the graph. Specializing
to staircase shapes, we show that the boolean numbers of the associated Ferrers
graphs are the Genocchi numbers of the second kind, and obtain a relation
between the Legendre-Stirling numbers and the Genocchi numbers of the second
kind. In another application, we compute the boolean number of a complete
bipartite graph, corresponding to a rectangular Ferrers shape, which is
expressed in terms of the Stirling numbers of the second kind. Finally, we
analyze the complexity of calculating the boolean number of a Ferrers graph
using these results and show that it is a significant improvement over
calculating by edge recursion.Comment: final version, to appear in the The Australasian Journal of
Combinatoric
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