Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.03017 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916987403239424 |
|---|---|
| author | Zapata, Cesar A. Ipanaque Borat, Ayse |
| author_facet | Zapata, Cesar A. Ipanaque Borat, Ayse |
| contents | We present the notion of facet-complexity, $\text{C}(\mathsf{L};\mathsf{K})$, for two simplicial complexes $\mathsf{L}$ and $\mathsf{K}$, along with basic results for this numerical invariant. This invariant $\text{C}(\mathsf{L};\mathsf{K})$ quantifies the \aspas{complexity} of the following question: When does there exist a facet simplicial map $\mathsf{L}\to \mathsf{K}$? A facet simplicial map is a simplicial map that preserves non-unitary facets. Likewise, we introduce the notion of injective facet-complexity, $\text{IC}(\mathsf{L};\mathsf{K})$. These invariants generalize the notion of (injective) hom-complexity between graphs, recently introduced by Zapata et al. We demonstrate a triangular inequality for (injective) facet-complexity and show that it is a simplicial complex invariant. Additionally, these invariants provide an obstruction to the existence of facet simplicial maps. We explore the sub-additivity of (injective) facet-complexity and we present a lower bound in terms of the chromatic number. Moreover, we provide an upper bound for $\mathrm{C}(\mathsf{L};\mathsf{H})$ in terms of the number of facets of $L$. Finally, we establish a formula for $\mathrm{IC}(\mathsf{L};\mathsf{K})$ when $\mathsf{L}$ is a pure simplicial complex and $K$ is a complete simplicial complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | (Injective) facet-complexity between simplicial complexes Zapata, Cesar A. Ipanaque Borat, Ayse Combinatorics We present the notion of facet-complexity, $\text{C}(\mathsf{L};\mathsf{K})$, for two simplicial complexes $\mathsf{L}$ and $\mathsf{K}$, along with basic results for this numerical invariant. This invariant $\text{C}(\mathsf{L};\mathsf{K})$ quantifies the \aspas{complexity} of the following question: When does there exist a facet simplicial map $\mathsf{L}\to \mathsf{K}$? A facet simplicial map is a simplicial map that preserves non-unitary facets. Likewise, we introduce the notion of injective facet-complexity, $\text{IC}(\mathsf{L};\mathsf{K})$. These invariants generalize the notion of (injective) hom-complexity between graphs, recently introduced by Zapata et al. We demonstrate a triangular inequality for (injective) facet-complexity and show that it is a simplicial complex invariant. Additionally, these invariants provide an obstruction to the existence of facet simplicial maps. We explore the sub-additivity of (injective) facet-complexity and we present a lower bound in terms of the chromatic number. Moreover, we provide an upper bound for $\mathrm{C}(\mathsf{L};\mathsf{H})$ in terms of the number of facets of $L$. Finally, we establish a formula for $\mathrm{IC}(\mathsf{L};\mathsf{K})$ when $\mathsf{L}$ is a pure simplicial complex and $K$ is a complete simplicial complex. |
| title | (Injective) facet-complexity between simplicial complexes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.03017 |