Geodesic
Biconed graphs, weighted forests, and \(h\)-vectors of matroid complexes
Preston Cranford ; Anton Dochtermann1 ; Evan Haithcock ; Joshua Marsh ; Suho Oh ; Anna Truman
1University of Texas at Austin
The electronic journal of combinatorics, Tome 28 (2021) no. 4
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A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids. We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.
DOI : 10.37236/9849
Classification : 05B35, 52B40, 13B25, 13H10
Mots-clés : graphic matroids, Stanley's conjecture

Preston Cranford    ; Anton Dochtermann  1   ; Evan Haithcock    ; Joshua Marsh    ; Suho Oh    ; Anna Truman 

1 University of Texas at Austin 
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     title = {Biconed graphs, weighted forests, and \(h\)-vectors of matroid complexes},
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     year = {2021},
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Preston Cranford; Anton Dochtermann; Evan Haithcock; Joshua Marsh; Suho Oh; Anna Truman. Biconed graphs, weighted forests, and \(h\)-vectors of matroid complexes. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/9849

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