844 research outputs found
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Small Ramsey Numbers
We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
A NEW GEOMETRIC MODEL AND METHODOLOGY FOR UNDERSTANDING PARSIMONIOUS SEVENTH-SONORITY PITCH-CLASS SPACE
Parsimonious voice leading is a term, first used by Richard Cohn, to describe non-diatonic motion among triads that will preserve as many common tones as possible, while limiting the distance traveled by the voice that does move to a tone or, better yet, a semitone. Some scholars have applied these principles to seventh chords, laying the groundwork for this study, which strives toward a reasonably comprehensive, usable model for musical analysis.
Rather than emphasizing mathematical proofs, as a number of approaches have done, this study relies on two- and three-dimensional geometric visualizations and spatial analogies to describe pitch-class and harmonic relationships. These geometric realizations are based on the organization of the neo-Riemannian Tonnetz, but they expand and apply the organizational principles of the Tonnetz to seventh sonorities. It allows for the descriptive “mapping” or prescriptive “navigation” of harmonic paths through a defined space.
The viability of the theoretical model is examined in analyses of passages from the repertoire of Frédéric Chopin. These passages exhibit a harmonic syntax that is often difficult to analyze as anything other than “tonally unstable” or “transitional.” This study seeks to analyze these passages in terms of what they are, rather than what they are not
Finding combinatorial structures
In this thesis we answer questions in two related areas of combinatorics:
Ramsey theory and asymptotic enumeration.
In Ramsey theory we introduce a new method for finding desired structures.
We find a new upper bound on the Ramsey number of a path against a kth
power of a path.
Using our new method and this result we obtain a new upper bound on the
Ramsey number of the kth power of a long cycle.
As a corollary we show that, while graphs on n vertices with maximum
degree k may in general have Ramsey numbers as large as ckn, if the stronger
restriction that the bandwidth should be at most k is given, then the Ramsey
numbers are bounded by the much smaller value.
We go on to attack an old conjecture of Lehel: by using our new method
we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new
method replaces their use of the Regularity Lemma, and allows us to prove
that for any n > 218000, whenever the edges of the complete graph on n
vertices are two-coloured there exist disjoint monochromatic cycles covering
all n vertices.
In asymptotic enumeration we examine first the class of bipartite graphs
with some forbidden induced subgraph H. We obtain some results for every
H, with special focus on the cases where the growth speed of the class is
factorial, and make some comments on a connection to clique-width. We
then move on to a detailed discussion of 2-SAT functions. We find the correct
asymptotic formula for the number of 2-SAT functions
on n variables (an improvement on a result of Bollob´as, Brightwell and
Leader [13], who found the dominant term in the exponent), the first error
term for this formula, and some bounds on smaller error terms. Finally
we obtain various expected values in the uniform model of random 2-SAT
functions
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