20,779 research outputs found
Independent Sets in Graphs with an Excluded Clique Minor
Let be a graph with vertices, with independence number , and
with with no -minor for some . It is proved that
On Multiplicative Sidon Sets
Fix integers with . A set is
\emph{-multiplicative} if for all . For all ,
we determine an -multiplicative set with maximum cardinality in ,
and conclude that the maximum density of an -multiplicative set is
. For , a set
is \emph{-multiplicative} if implies and for
all and , and . For and
coprime, we give an O(1) time algorithm to approximate the maximum density of
an -multiplicative set to arbitrary given precision
The "Double Sense" of Fichte's Philosophical Language - Some Critical Reflections on the Cambridge Companion to Fichte
The principal thesis in this review-essay is that the key linguistic terms in Fichte’s Wissenschaftslehre especially have two main meanings that appear at first sight to be almost in contradiction or opposed to each other. The reader of Fichte therefore has to work hard to overcome any apparent conflicts in the “double sense” of his philosophical terminology. Accordingly, I argue that Fichte’s linguistic method and use of language should be seen as part of his chief philosophical method of synthesis, where we have to carry out a similar procedure and attempt to reconcile opposites using the power of the imagination. This thesis is put forward by means of a number of practical examples and in the context of some critical reflections on the recently published Cambridge Companion to Fichte, eds. David James and Günter Zöller. Review essay published in Volume 15 (December, 2017) of the journal Revista de Estud(i)os sobre Fichte (ed. Emiliano Acosta)
Colouring the Square of the Cartesian Product of Trees
We prove upper and lower bounds on the chromatic number of the square of the
cartesian product of trees. The bounds are equal if each tree has even maximum
degree
On Tree-Partition-Width
A \emph{tree-partition} of a graph is a proper partition of its vertex
set into `bags', such that identifying the vertices in each bag produces a
forest. The \emph{tree-partition-width} of is the minimum number of
vertices in a bag in a tree-partition of . An anonymous referee of the paper
by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph
with tree-width and maximum degree has
tree-partition-width at most . We prove that this bound is within a
constant factor of optimal. In particular, for all and for all
sufficiently large , we construct a graph with tree-width , maximum
degree , and tree-partition-width at least (\eighth-\epsilon)k\Delta.
Moreover, we slightly improve the upper bound to
without the restriction that
Drawing a Graph in a Hypercube
A -dimensional hypercube drawing of a graph represents the vertices by
distinct points in , such that the line-segments representing the
edges do not cross. We study lower and upper bounds on the minimum number of
dimensions in hypercube drawing of a given graph. This parameter turns out to
be related to Sidon sets and antimagic injections.Comment: Submitte
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