1,988 research outputs found
Proper orientation of cacti
An orientation of a graph is proper if two adjacent vertices have different indegrees. We prove that every cactus admits a proper orientation with maximum indegree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum indegree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum indegree at most 6 and that this bound can also be attained
Proper orientation of cacti
International audienceAn orientation of a graph G is proper if two adjacent vertices have different in-degrees. The proper-orientation number − → χ (G) of a graph G is the minimum maximum in-degree of a proper orientation of G. In [1], the authors ask whether the proper orientation number of a planar graph is bounded. We prove that every cactus admits a proper orientation with maximum in-degree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum in-degree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum in-degree at most 6 and that this bound can also be attained
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
Arc Operads and Arc Algebras
Several topological and homological operads based on families of projectively
weighted arcs in bounded surfaces are introduced and studied. The spaces
underlying the basic operad are identified with open subsets of a
compactification due to Penner of a space closely related to Riemann's moduli
space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras,
where the entire BV structure is realized simplicially. Furthermore, our basic
operad contains the cacti operad up to homotopy, and it similarly acts on the
loop space of any topological space. New operad structures on the circle are
classified and combined with the basic operad to produce geometrically natural
extensions of the algebraic structure of BV algebras, which are also computed.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper15.abs.htm
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