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Critical Pebbling Numbers of Graphs
We define three new pebbling parameters of a connected graph , the -,
-, and -critical pebbling numbers. Together with the pebbling number, the
optimal pebbling number, the number of vertices and the diameter of the
graph, this yields 7 graph parameters. We determine the relationships between
these parameters. We investigate properties of the -critical pebbling
number, and distinguish between greedy graphs, thrifty graphs, and graphs for
which the -critical pebbling number is .Comment: 26 page
On several varieties of cacti and their relations
Motivated by string topology and the arc operad, we introduce the notion of
quasi-operads and consider four (quasi)-operads which are different varieties
of the operad of cacti. These are cacti without local zeros (or spines) and
cacti proper as well as both varieties with fixed constant size one of the
constituting loops. Using the recognition principle of Fiedorowicz, we prove
that spineless cacti are equivalent as operads to the little discs operad. It
turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and
Fiedorowicz' braided operad structure are given by the same explicit chains. We
also prove that spineless cacti and cacti are homotopy equivalent to their
normalized versions as quasi-operads by showing that both types of cacti are
semi-direct products of the quasi-operad of their normalized versions with a
re-scaling operad based on R>0. Furthermore, we introduce the notion of
bi-crossed products of quasi-operads and show that the cacti proper are a
bi-crossed product of the operad of cacti without spines and the operad based
on the monoid given by the circle group S^1. We also prove that this particular
bi-crossed operad product is homotopy equivalent to the semi-direct product of
the spineless cacti with the group S^1. This implies that cacti are equivalent
to the framed little discs operad. These results lead to new CW models for the
little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm
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