4,016 research outputs found
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
Vertex arboricity of triangle-free graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order
New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix
The purpose of this article is to improve existing lower bounds on the
chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency
matrix sorted in non-increasing order.
First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / -
sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower
bound which only involves the maximum and minimum eigenvalues, i.e., the case
. We provide several examples for which the new bound exceeds the {\sc
Hoffman} lower bound.
Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^-
are the sums of the squares of positive and negative eigenvalues, respectively.
To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We
show that the conjectured lower bound is tight for several families of graphs.
We also performed various searches for a counter-example, but none was found.
Our proofs rely on a new technique of converting the adjacency matrix into
the zero matrix by conjugating with unitary matrices and use majorization of
spectra of self-adjoint matrices.
We also show that the above bounds are actually lower bounds on the
normalized orthogonal rank of a graph, which is always less than or equal to
the chromatic number. The normalized orthogonal rank is the minimum dimension
making it possible to assign vectors with entries of modulus one to the
vertices such that two such vectors are orthogonal if the corresponding
vertices are connected.
All these bounds are also valid when we replace the adjacency matrix A by W *
A where W is an arbitrary self-adjoint matrix and * denotes the Schur product,
that is, entrywise product of W and A
The Parameterised Complexity of List Problems on Graphs of Bounded Treewidth
We consider the parameterised complexity of several list problems on graphs,
with parameter treewidth or pathwidth. In particular, we show that List Edge
Chromatic Number and List Total Chromatic Number are fixed parameter tractable,
parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even
parameterised by pathwidth. These results resolve two open questions of
Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).Comment: Author final version, to appear in Information and Computation.
Changes from previous version include improved literature references and
restructured proof in Section
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
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
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