The homology of the cyclic coloring complex of simple graphs

Title:
The homology of the cyclic coloring complex of simple graphs
Authors:
Rundell, Sarah
Citation:
Crown, S. (2009). The homology of the cyclic coloring complex of simple graphs. Journal of Combinatorial Theory, Series A, 116(3), 595-612.
Publisher:
Journal of Combinatorial Theory
DATE ISSUED:
2009
PERMANENT LINK:
http://hdl.handle.net/2374.DEN/5023; http://hdl.handle.net/2374
Type:
Article
Language:
en_US
Description:
Let G be a simple graph on n vertices, and let I.sub.G(I') denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, [DELTA](G), and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of (n-3)rd homology group of [DELTA](G) is equal to (n-(r+1)) plus 1/r!| I.sub.G.sup.r(0)|, where I.sub.G.sup.r is the rth derivative of I.sub.G(I'). We also define a complex [DELTA](G).sup.C, whose r-faces consist of all ordered set partitions [B.sub.1,...,B.sub.r+2] where none of the B.sub.i contain an edge of G and where 1[member of]B.sub.1. We compute the dimensions of the homology groups of this complex, and as a result, obtain the dimensions of the multilinear parts of the cyclic homology groups of C[x.sub.1,...,x.sub.n]/{x.sub.ix.sub.j|ij is an edge of G}. We show that when G is a connected graph, the homology of [DELTA](G).sup.C has nonzero homology only in dimension n-2, and the dimension of this homology group is | I.sub.G.sup.'(0)|. In this case, we provide a bijection between a set of homology representatives of [DELTA](G).sup.C and the acyclic orientations of G with a unique source at v, a vertex of G.
ISSN:
00973165
Appears in Collections:
Faculty Publications

Full metadata record

DC FieldValue Language
dc.contributor.authorRundell, Sarahen
dc.date.accessioned2013-01-02T18:10:34Zen
dc.date.accessioned2013-12-18T21:06:14Z-
dc.date.available2013-01-02T18:10:34Zen
dc.date.available2013-12-18T21:06:14Z-
dc.date.created2009en
dc.date.issued2009en
dc.identifier.citationCrown, S. (2009). The homology of the cyclic coloring complex of simple graphs. Journal of Combinatorial Theory, Series A, 116(3), 595-612.en_US
dc.identifier.issn00973165en
dc.identifier.urihttp://hdl.handle.net/2374.DEN/5023en
dc.identifier.urihttp://hdl.handle.net/2374-
dc.descriptionLet G be a simple graph on n vertices, and let I.sub.G(I') denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, [DELTA](G), and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of (n-3)rd homology group of [DELTA](G) is equal to (n-(r+1)) plus 1/r!| I.sub.G.sup.r(0)|, where I.sub.G.sup.r is the rth derivative of I.sub.G(I'). We also define a complex [DELTA](G).sup.C, whose r-faces consist of all ordered set partitions [B.sub.1,...,B.sub.r+2] where none of the B.sub.i contain an edge of G and where 1[member of]B.sub.1. We compute the dimensions of the homology groups of this complex, and as a result, obtain the dimensions of the multilinear parts of the cyclic homology groups of C[x.sub.1,...,x.sub.n]/{x.sub.ix.sub.j|ij is an edge of G}. We show that when G is a connected graph, the homology of [DELTA](G).sup.C has nonzero homology only in dimension n-2, and the dimension of this homology group is | I.sub.G.sup.'(0)|. In this case, we provide a bijection between a set of homology representatives of [DELTA](G).sup.C and the acyclic orientations of G with a unique source at v, a vertex of G.en_US
dc.language.isoen_USen_US
dc.publisherJournal of Combinatorial Theoryen_US
dc.relation.ispartofFaculty Publicationsen_US
dc.titleThe homology of the cyclic coloring complex of simple graphsen_US
dc.typeArticleen_US
dc.contributor.institutionDenison Universityen_US
dc.date.digitized2013-01-02en
dc.contributor.repositoryDenison Resource Commonsen_US
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