Asymmetric results about graph homomorphisms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910911503007744 |
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| author | Gishboliner, Lior Hurley, Eoin Wigderson, Yuval |
| author_facet | Gishboliner, Lior Hurley, Eoin Wigderson, Yuval |
| contents | Many important results in extremal graph theory can be roughly summarised as "if a triangle-free graph $G$ has certain properties, then it has a homomorphism to a triangle-free graph $Γ$ of bounded size". For example, bounds on homomorphism thresholds give such a statement if $G$ has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all $G$, if one weakens the notion of homomorphism appropriately.
In this paper, we study asymmetric versions of these results, where the assumptions on $G$ and $Γ$ need not match. For example, we prove that if $G$ is a graph with odd girth at least $9$ and minimum degree at least $δ|G|$, then $G$ is homomorphic to a triangle-free graph whose size depends only on $δ$. Moreover, the odd girth assumption can be weakened to odd girth at least $7$ if $G$ has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang et al.
We also prove that in the asymmetric approximate homomorphism theorem, the bounds exhibit a rather surprising ``double phase transition'': the bounds are super-exponential if $G$ is only assumed to be triangle-free, they become exponential if $G$ is assumed to have odd girth $7$ or $9$, and become linear if $G$ has odd girth at least $11$.
Our proofs use a wide variety of techniques, including entropy arguments, the Frieze--Kannan weak regularity lemma, properties of the generalised Mycielskian construction, and recent work on abundance and the asymmetric removal lemma. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_20278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymmetric results about graph homomorphisms Gishboliner, Lior Hurley, Eoin Wigderson, Yuval Combinatorics Many important results in extremal graph theory can be roughly summarised as "if a triangle-free graph $G$ has certain properties, then it has a homomorphism to a triangle-free graph $Γ$ of bounded size". For example, bounds on homomorphism thresholds give such a statement if $G$ has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all $G$, if one weakens the notion of homomorphism appropriately. In this paper, we study asymmetric versions of these results, where the assumptions on $G$ and $Γ$ need not match. For example, we prove that if $G$ is a graph with odd girth at least $9$ and minimum degree at least $δ|G|$, then $G$ is homomorphic to a triangle-free graph whose size depends only on $δ$. Moreover, the odd girth assumption can be weakened to odd girth at least $7$ if $G$ has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang et al. We also prove that in the asymmetric approximate homomorphism theorem, the bounds exhibit a rather surprising ``double phase transition'': the bounds are super-exponential if $G$ is only assumed to be triangle-free, they become exponential if $G$ is assumed to have odd girth $7$ or $9$, and become linear if $G$ has odd girth at least $11$. Our proofs use a wide variety of techniques, including entropy arguments, the Frieze--Kannan weak regularity lemma, properties of the generalised Mycielskian construction, and recent work on abundance and the asymmetric removal lemma. |
| title | Asymmetric results about graph homomorphisms |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.20278 |