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Dirac's Theorem on Simplicial Matroids
Raul Cordovil1, Manoel Lemos2,Cl\'audia Linhares Sales3
1Departamento de Matem\'atica, Instituto Superior T\'ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
cordovil@math.ist.utl.pt
2Departamento de Matem\'atica, Universidade Federal de Pernambuco, Recife, Pernambuco, CEP 50740-540, Brasil
manoel@dmat.ufpe.br
3MDCC, Departamento de Computa\c c\~ao, Universidade Federal do Cear\'a - UFC, Campus do Pici, Bloco 910, Fortaleza, CE, Brasil
linhares@lia.ufc.br
Annals of Combinatorics 13 (1) pp.53-63 March, 2009
AMS Subject Classification: 05B35; 05C17
Abstract:
We introduce the notion of $k$-hyper\-clique complexes, i.e., the largest simplicial complexes on the set $[n]$ with a fixed $k$-skeleton. These simplicial complexes are a higher-dimen\-sio\-nal analogue of clique (or flag) complexes (case $k=2$) and they are a rich new class of simplicial complexes. We show that Dirac's theorem on chordal graphs has a higher-dimen\-sio\-nal analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and $k$-hyper\-clique complexes. We prove also a higher-dimen\-sio\-nal analogue of Stanley's reformulation of Dirac's theorem on chordal graphs.
Keywords: clique (flag) complexes, Dirac's theorem on chordal graphs, simplicial matroids, $k$-hyperclique complexes, Helly dual $k$-property, strong triangulable simplicial matroids

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