Planar graphs as minimal resolutions of trivariate monomial ideals
type: Item
instance of: scholarly article
title: Planar graphs as minimal resolutions of trivariate monomial ideals
zbMATH Open document ID: 0989.05026
author: Ezra Miller
published in: Documenta Mathematica
publication date: 2002-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/121888
full work available at URL: 03.html
review text: Summary: We introduce the notion of rigid embedding in a grid surface, a new kind of plane drawing for simple triconnected planar graphs. Rigid embeddings provide methods to (1)~find well-structured (cellular, here) minimal free resolutions for arbitrary monomial ideals in three variables; (2)~strengthen the Brightwell--Trotter bound on the order dimension of triconnected planar maps by giving a geometric reformulation; and (3)~generalize Schnyder's angle coloring of planar triangulations to arbitrary triconnected planar maps via geometry. The notion of rigid embedding is stable under duality for planar maps, and has certain uniqueness properties.
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 68R10
Mathematics Subject Classification ID: 52Cxx
zbMATH DE Number: 1748377
zbMATH Keywords: free resolution
zbMATH Keywords: order dimension
zbMATH Keywords: (rigid) geodesic embedding
zbMATH Keywords: planar graphs
zbMATH Keywords: monomial idels
zbMATH Keywords: planar maps
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1748377