1,891 research outputs found
Cover pebbling numbers and bounds for certain families of graphs
Given a configuration of pebbles on the vertices of a graph, a pebbling move
is defined by removing two pebbles from some vertex and placing one pebble on
an adjacent vertex. The cover pebbling number of a graph, gamma(G), is the
smallest number of pebbles such that through a sequence of pebbling moves, a
pebble can eventually be placed on every vertex simultaneously, no matter how
the pebbles are initially distributed. The cover pebbling number for complete
multipartite graphs and wheel graphs is determined. We also prove a sharp bound
for gamma(G) given the diameter and number of vertices of G.Comment: 10 pages, 1 figure, submitted to Discrete Mathematic
Domination Cover Pebbling: Structural Results
This paper continues the results of "Domination Cover Pebbling: Graph
Families." An almost sharp bound for the domination cover pebbling (DCP) number
for graphs G with specified diameter has been computed. For graphs of diameter
two, a bound for the ratio between the cover pebbling number of G and the DCP
number of G has been computed. A variant of domination cover pebbling, called
subversion DCP is introducted, and preliminary results are discussed.Comment: 15 page
Non-Simplicial Nerves for Two-Dimensional Categorical Structures
The most natural notion of a simplicial nerve for a (weak) bicategory was
given by Duskin, who showed that a simplicial set is isomorphic to the nerve of
a -category (i.e. a bicategory with invertible -morphisms) if and
only if it is a quasicategory which has unique fillers for inner horns of
dimension and greater. Using Duskin's technique, we show how his nerve
applies to -category functors, making it a fully faithful inclusion of
-categories into simplicial sets. Then we consider analogues of this
extension of Duskin's result for several different two-dimensional categorical
structures, defining and analysing nerves valued in presheaf categories based
on , on Segal's category , and Joyal's category .
In each case, our nerves yield exactly those presheaves meeting a certain
"horn-filling" condition, with unique fillers for high-dimensional horns.
Generalizing our definitions to higher dimensions and relaxing this uniqueness
condition, we get proposed models for several different kinds
higher-categorical structures, with each of these models closely analogous to
quasicategories. Of particular interest, we conjecture that our "inner-Kan
-sets'' are a combinatorial model for symmetric monoidal
-categories, i.e. -spaces.
This is a version of the author's Ph.D. dissertation, completed 2013 at the
University of California, Berkeley. Minor corrections and changes are included.Comment: 247 page. Ph.D. Dissertation (2013
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