On graphs embeddable in a layer of a hypercube and their extremal numbers
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| Format: | Preprint |
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2023
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| author | Axenovich, Maria Martin, Ryan R. Winter, Christian |
| author_facet | Axenovich, Maria Martin, Ryan R. Winter, Christian |
| contents | A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is equal to a positive proportion of the number of edges in $Q_n$, $H$ is said to have positive Turán density in a hypercube; otherwise it has zero Turán density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has positive or zero Turán density remains a widely open question for general $H$.
In this paper we focus on layered graphs, i.e., graphs that are contained in an edge-layer of some hypercube. Graphs $H$ that are not layered have positive Turán density because one can form an $H$-free subgraph of $Q_n$ consisting of edges of every other layer. For example, a $4$-cycle is not layered and has positive Turán density.
However, in general it is not obvious what properties layered graphs have. We give a characterisation of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least $12$ and of length $8$. However, we prove that there are cubical graphs of girth $8$ that are not layered and thus having positive Turán density. The cycle of length $10$ remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that $ex(Q_n, C_{10})= Ω(n2^n/ \log^a n)$, for a constant $a$, showing that the extremal number for a $10$-cycle behaves differently from any other cycle of zero Turán density. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_15529 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On graphs embeddable in a layer of a hypercube and their extremal numbers Axenovich, Maria Martin, Ryan R. Winter, Christian Combinatorics 05 A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is equal to a positive proportion of the number of edges in $Q_n$, $H$ is said to have positive Turán density in a hypercube; otherwise it has zero Turán density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has positive or zero Turán density remains a widely open question for general $H$. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge-layer of some hypercube. Graphs $H$ that are not layered have positive Turán density because one can form an $H$-free subgraph of $Q_n$ consisting of edges of every other layer. For example, a $4$-cycle is not layered and has positive Turán density. However, in general it is not obvious what properties layered graphs have. We give a characterisation of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least $12$ and of length $8$. However, we prove that there are cubical graphs of girth $8$ that are not layered and thus having positive Turán density. The cycle of length $10$ remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that $ex(Q_n, C_{10})= Ω(n2^n/ \log^a n)$, for a constant $a$, showing that the extremal number for a $10$-cycle behaves differently from any other cycle of zero Turán density. |
| title | On graphs embeddable in a layer of a hypercube and their extremal numbers |
| topic | Combinatorics 05 |
| url | https://arxiv.org/abs/2303.15529 |