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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.05902 |
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| _version_ | 1866913381799165952 |
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| author | Fox, Jacob Nenadov, Rajko Pham, Huy Tuan |
| author_facet | Fox, Jacob Nenadov, Rajko Pham, Huy Tuan |
| contents | We initiate the systematic study of the following Turán-type question. Suppose $Γ$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$ and $c > 0$. What is the largest number of edges in a subgraph $G \subseteq Γ$ which does not contain a fixed graph $H$ as an induced subgraph or, more generally, which belongs to a hereditary property $\mathcal{P}$? This provides a common generalization of two recently studied cases, namely $Γ$ being a (pseudo-)random graph and a graph without a large complete bipartite subgraph. We focus on the interesting case where $H$ is a bipartite graph.
We determine the answer up to a constant factor with respect to $n$ and $t$, for certain bipartite $H$ and for $Γ$ either a dense random graph or a Paley graph with a square number of vertices. In particular, our bounds match if $H$ is a tree, or if one part of $H$ has $d$ vertices complete to the other part, all other vertices in that part have degree at most $d$, and the other part has sufficiently many vertices. As applications of the latter result, we answer a question of Alon, Krivelevich, and Samotij on the largest subgraph with a hereditary property which misses a bipartite graph, and determine up to a constant factor the largest number of edges in a string subgraph of $Γ$. The proofs are based on a variant of the dependent random choice and a novel approach for finding induced copies by inductively defining probability distributions supported on induced copies of smaller subgraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The largest subgraph without a forbidden induced subgraph Fox, Jacob Nenadov, Rajko Pham, Huy Tuan Combinatorics Probability We initiate the systematic study of the following Turán-type question. Suppose $Γ$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$ and $c > 0$. What is the largest number of edges in a subgraph $G \subseteq Γ$ which does not contain a fixed graph $H$ as an induced subgraph or, more generally, which belongs to a hereditary property $\mathcal{P}$? This provides a common generalization of two recently studied cases, namely $Γ$ being a (pseudo-)random graph and a graph without a large complete bipartite subgraph. We focus on the interesting case where $H$ is a bipartite graph. We determine the answer up to a constant factor with respect to $n$ and $t$, for certain bipartite $H$ and for $Γ$ either a dense random graph or a Paley graph with a square number of vertices. In particular, our bounds match if $H$ is a tree, or if one part of $H$ has $d$ vertices complete to the other part, all other vertices in that part have degree at most $d$, and the other part has sufficiently many vertices. As applications of the latter result, we answer a question of Alon, Krivelevich, and Samotij on the largest subgraph with a hereditary property which misses a bipartite graph, and determine up to a constant factor the largest number of edges in a string subgraph of $Γ$. The proofs are based on a variant of the dependent random choice and a novel approach for finding induced copies by inductively defining probability distributions supported on induced copies of smaller subgraphs. |
| title | The largest subgraph without a forbidden induced subgraph |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2405.05902 |