Regular subgraphs at every density
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912730077724672 |
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| author | Chakraborti, Debsoumya Janzer, Oliver Methuku, Abhishek Montgomery, Richard |
| author_facet | Chakraborti, Debsoumya Janzer, Oliver Methuku, Abhishek Montgomery, Richard |
| contents | In 1975, Erdős and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any $n$-vertex graph with no $r$-regular subgraph has at most $C_r n \log \log n$ edges, matching an earlier lower bound by Pyber, Rödl and Szemerédi and thereby resolving the Erdős-Sauer problem up to a constant depending on $r$. We prove that every $n$-vertex graph without an $r$-regular subgraph has at most $Cr^2 n \log \log n$ edges. This bound is tight up to the value of $C$ for $n\geq n_0(r)$ and hence resolves the Erdős-Sauer problem up to an absolute constant.
Moreover, we obtain similarly tight results for the whole range of possible values of $r$ (i.e., not just when $r$ is a constant), apart from a small error term at a transition point near $r\approx \log n$, where, perhaps surprisingly, the answer changes. More specifically, we show that every $n$-vertex graph with average degree at least $\min(Cr\log(n/r),Cr^2 \log\log n)$ contains an $r$-regular subgraph. The bound $Cr\log(n/r)$ is tight for $r\geq \log n$, while the bound $Cr^2 \log \log n$ is tight for $r<(\log n)^{1-Ω(1)}$. These results resolve a problem of Rödl and Wysocka from 1997 for almost all values of $r$.
Among other tools, we develop a novel random process that efficiently finds a very nearly regular subgraph in any almost-regular graph. A key step in our proof uses this novel random process to show that every $K$-almost-regular graph with average degree $d$ contains an $r$-regular subgraph for some $r=Ω_K(d)$, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11785 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular subgraphs at every density Chakraborti, Debsoumya Janzer, Oliver Methuku, Abhishek Montgomery, Richard Combinatorics In 1975, Erdős and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any $n$-vertex graph with no $r$-regular subgraph has at most $C_r n \log \log n$ edges, matching an earlier lower bound by Pyber, Rödl and Szemerédi and thereby resolving the Erdős-Sauer problem up to a constant depending on $r$. We prove that every $n$-vertex graph without an $r$-regular subgraph has at most $Cr^2 n \log \log n$ edges. This bound is tight up to the value of $C$ for $n\geq n_0(r)$ and hence resolves the Erdős-Sauer problem up to an absolute constant. Moreover, we obtain similarly tight results for the whole range of possible values of $r$ (i.e., not just when $r$ is a constant), apart from a small error term at a transition point near $r\approx \log n$, where, perhaps surprisingly, the answer changes. More specifically, we show that every $n$-vertex graph with average degree at least $\min(Cr\log(n/r),Cr^2 \log\log n)$ contains an $r$-regular subgraph. The bound $Cr\log(n/r)$ is tight for $r\geq \log n$, while the bound $Cr^2 \log \log n$ is tight for $r<(\log n)^{1-Ω(1)}$. These results resolve a problem of Rödl and Wysocka from 1997 for almost all values of $r$. Among other tools, we develop a novel random process that efficiently finds a very nearly regular subgraph in any almost-regular graph. A key step in our proof uses this novel random process to show that every $K$-almost-regular graph with average degree $d$ contains an $r$-regular subgraph for some $r=Ω_K(d)$, which is of independent interest. |
| title | Regular subgraphs at every density |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.11785 |