Planar Graphs and the Koszul Algebra Structure for Trivariate Monomial Ideals

2014 ◽  
pp. 277-297
Author(s):  
Jared L. Painter
Keyword(s):  
Planar Graphs ◽  
Monomial Ideals ◽  
Algebra Structure ◽  
Koszul Algebra

scholarly journals DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

2018 ◽  
Vol 238 ◽  
pp. 47-85 ◽  
Author(s):  
AMANDA CROLL ◽  
ROGER DELLACA ◽  
ANJAN GUPTA ◽  
JUSTIN HOFFMEIER ◽  
VIVEK MUKUNDAN ◽  
...  
Keyword(s):  
Poincaré Series ◽  
Koszul Complex ◽  
Local Rings ◽  
Common Denominator ◽  
Algebra Structure ◽  
Koszul Algebra ◽  
Multiplicative Structure ◽  
Poincare Series ◽  
Koszul Homology ◽  
Finitely Generated Modules

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

A Space-Efficient Separator Algorithm for Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 1007-1016 ◽  
Author(s):  
Ryo ASHIDA ◽  
Sebastian KUHNERT ◽  
Osamu WATANABE
Keyword(s):  
Planar Graphs

Note on (semi-)proper orientation of some triangulated planar graphs

2021 ◽  
Vol 392 ◽  
pp. 125723
Author(s):  
Ruijuan Gu ◽  
Hui Lei ◽  
Yulai Ma ◽  
Zhenyu Taoqiu
Keyword(s):  
Planar Graphs ◽  
Proper Orientation

scholarly journals Distributed Dominating Set Approximations beyond Planar Graphs

2019 ◽  
Vol 15 (3) ◽  
pp. 1-18 ◽  
Author(s):  
Saeed Akhoondian Amiri ◽  
Stefan Schmid ◽  
Sebastian Siebertz
Keyword(s):  
Planar Graphs ◽  
Dominating Set

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

scholarly journals Blackbox computation of A ∞-algebras

2010 ◽  
Vol 17 (2) ◽  
pp. 391-404
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Homology Theory ◽  
Algebra Structure ◽  
Finite Amount ◽  
Computational Work ◽  
Dg Algebra ◽  
Dg Algebras

Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

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