Barile-Macchia Resolutions and the closed neighborhood ideal
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arXiv
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| Format: | Preprint |
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2025
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| author | Joseph, Ajay P. Roy, Amit Singh, Anurag |
| author_facet | Joseph, Ajay P. Roy, Amit Singh, Anurag |
| contents | We investigate the minimal free resolutions of closed neighborhood ideals of graphs within the framework of Barile-Macchia (BM) resolutions. We show that for any tree $T$, the closed neighborhood ideal $NI(T)$ is bridge-friendly, and hence its BM resolution is minimal. The combinatorial structure of trees further allows us to construct a maximal critical cell of size $α(T)$, leading to the equality $\operatorname{pd}(R/NI(T)) = α(T)$, where $α(T)$ denotes the independence number of $T$ and $\operatorname{pd}$ is the projective dimension. Using Betti splitting techniques, we also obtain explicit formulas for the graded Betti numbers of $NI(P_n)$, where $P_n$ is the path graph on $n$ vertices. Finally, we make some observations on the bridge-friendly condition of the closed neighborhood ideals of chordal and bipartite graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_22657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Barile-Macchia Resolutions and the closed neighborhood ideal Joseph, Ajay P. Roy, Amit Singh, Anurag Commutative Algebra Combinatorics 13D02, 13F55, 05E40, 05C05 We investigate the minimal free resolutions of closed neighborhood ideals of graphs within the framework of Barile-Macchia (BM) resolutions. We show that for any tree $T$, the closed neighborhood ideal $NI(T)$ is bridge-friendly, and hence its BM resolution is minimal. The combinatorial structure of trees further allows us to construct a maximal critical cell of size $α(T)$, leading to the equality $\operatorname{pd}(R/NI(T)) = α(T)$, where $α(T)$ denotes the independence number of $T$ and $\operatorname{pd}$ is the projective dimension. Using Betti splitting techniques, we also obtain explicit formulas for the graded Betti numbers of $NI(P_n)$, where $P_n$ is the path graph on $n$ vertices. Finally, we make some observations on the bridge-friendly condition of the closed neighborhood ideals of chordal and bipartite graphs. |
| title | Barile-Macchia Resolutions and the closed neighborhood ideal |
| topic | Commutative Algebra Combinatorics 13D02, 13F55, 05E40, 05C05 |
| url | https://arxiv.org/abs/2511.22657 |