Geodesic
Simplicial complexes with many facets are vertex decomposable
Anton Dochtermann1 ; Ritika Nair2 ; Jay Schweig3 ; Adam Van Tuyl4 ; Russ Woodroofe5
1Texas State University
2University of Kansas
3Oklahoma State University
4McMaster University
5University of Primorska
The electronic journal of combinatorics, Tome 31 (2024) no. 4
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Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$, then $\Delta$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $\Delta$ is vertex decomposable. We give examples to show that this bound is optimal, and that the conclusion cannot be strengthened to the class of matroids or shifted complexes. We explore an application to Simon's Conjecture and discuss connections to other results from the literature.
DOI : 10.37236/12984
Classification : 05E40, 05E45, 13F55
Mots-clés : simplicial complexes, facets, vertex decomposable, Simon's conjecture

Anton Dochtermann  1   ; Ritika Nair  2   ; Jay Schweig  3   ; Adam Van Tuyl  4   ; Russ Woodroofe  5

1 Texas State University
2 University of Kansas
3 Oklahoma State University
4 McMaster University
5 University of Primorska 
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     title = {Simplicial complexes with many facets are vertex decomposable},
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     year = {2024},
     volume = {31},
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     url = {http://geodesic.mathdoc.fr/articles/10.37236/12984/}
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Anton Dochtermann; Ritika Nair; Jay Schweig; Adam Van Tuyl; Russ Woodroofe. Simplicial complexes with many facets are vertex decomposable. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12984

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