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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2411.16547 |
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| _version_ | 1866912463845326848 |
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| author | Zapata, Cesar A. Ipanaque Enciso, Josué A. Aguirre Ramos, Wilman Francisco Cuba |
| author_facet | Zapata, Cesar A. Ipanaque Enciso, Josué A. Aguirre Ramos, Wilman Francisco Cuba |
| contents | We present the notion of hom-complexity, $\text{C}(G;H)$, for two graphs $G$ and $H$, along with basic results for this numerical invariant. This invariant $\text{C}(G;H)$ is a number that measures the \aspas{complexity} of the question: when is there a homomorphism $G\to H$? More precisely, $\text{C}(G;H)$ is the least positive integer $k$ such that there are $k$ different subgraphs $G_j$ of $G$ such that $G=G_1\cup\cdots\cup G_k$, and for each $G_j$, there is a homomorphism $G_j\to H$. Likewise, we introduce the notion of injective hom-complexity, $\text{IC}(G;H)$. The (injective) hom-complexity is a graph invariant. Additionally, these invariants can be used to show the nonexistence of homomorphisms. We explore the sub-additivity of (injective) hom-complexity and study products.
We describe bounds for the hom-complexity in terms of chromatic number $χ$ and clique number $ω$. We provide the formula \[\text{C}(G;H)=\lceil\log_{χ(H)}χ(G)\rceil\] whenever $ω(H)=χ(H)$. For example, we obtain $\text{C}(G;K_\ell)=\lceil\log_{\ell}χ(G)\rceil$. Moreover, we discuss a connection between the (injective) hom-complexity and several well-known covering numbers. For instance, we provide a lower bound for the clique covering number in terms of the injective hom-complexity. Additionally, we show that the hom-complexity $\mathrm{C}(G;K_{\ell})$ coincides with the $\ell$-particity $β_\ell(G)$ of $G$, and the hom-complexity $\mathrm{C}(K_n;K_{2})$ coincides with the bipartite dimension $\mathrm{d}(K_n)$ of $K_n$. As a consequence, we recover the well-known formulas $β_\ell(G)=\lceil\log_{\ell}χ(G)\rceil$ and $\mathrm{d}(K_n)=\lceil\log_{2}n\rceil$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16547 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | (Injective) hom-complexity between graphs Zapata, Cesar A. Ipanaque Enciso, Josué A. Aguirre Ramos, Wilman Francisco Cuba Combinatorics We present the notion of hom-complexity, $\text{C}(G;H)$, for two graphs $G$ and $H$, along with basic results for this numerical invariant. This invariant $\text{C}(G;H)$ is a number that measures the \aspas{complexity} of the question: when is there a homomorphism $G\to H$? More precisely, $\text{C}(G;H)$ is the least positive integer $k$ such that there are $k$ different subgraphs $G_j$ of $G$ such that $G=G_1\cup\cdots\cup G_k$, and for each $G_j$, there is a homomorphism $G_j\to H$. Likewise, we introduce the notion of injective hom-complexity, $\text{IC}(G;H)$. The (injective) hom-complexity is a graph invariant. Additionally, these invariants can be used to show the nonexistence of homomorphisms. We explore the sub-additivity of (injective) hom-complexity and study products. We describe bounds for the hom-complexity in terms of chromatic number $χ$ and clique number $ω$. We provide the formula \[\text{C}(G;H)=\lceil\log_{χ(H)}χ(G)\rceil\] whenever $ω(H)=χ(H)$. For example, we obtain $\text{C}(G;K_\ell)=\lceil\log_{\ell}χ(G)\rceil$. Moreover, we discuss a connection between the (injective) hom-complexity and several well-known covering numbers. For instance, we provide a lower bound for the clique covering number in terms of the injective hom-complexity. Additionally, we show that the hom-complexity $\mathrm{C}(G;K_{\ell})$ coincides with the $\ell$-particity $β_\ell(G)$ of $G$, and the hom-complexity $\mathrm{C}(K_n;K_{2})$ coincides with the bipartite dimension $\mathrm{d}(K_n)$ of $K_n$. As a consequence, we recover the well-known formulas $β_\ell(G)=\lceil\log_{\ell}χ(G)\rceil$ and $\mathrm{d}(K_n)=\lceil\log_{2}n\rceil$. |
| title | (Injective) hom-complexity between graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.16547 |