Geodesic
Planar graphs as minimal resolutions of trivariate monomial ideals
Documenta mathematica, Tome 7 (2002), pp. 43-90
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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.
DOI : 10.4171/dm/117
Classification : 05C10, 06A07, 13D02, 13F55, 52Cxx, 68R10
Mots-clés : planar graph, monomial ideal, free resolution, order dimension, (rigid) geodesic embedding 
@article{10_4171_dm_117,
     author = {Ezra Miller},
     title = {Planar graphs as minimal resolutions of trivariate monomial ideals},
     journal = {Documenta mathematica},
     pages = {43--90},
     year = {2002},
     volume = {7},
     doi = {10.4171/dm/117},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/117/}
}
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Ezra Miller. Planar graphs as minimal resolutions of trivariate monomial ideals. Documenta mathematica, Tome 7 (2002), pp. 43-90. doi: 10.4171/dm/117

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