Solution to a problem of Erdős on the chromatic index of hypergraphs with bounded codegree
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| Format: | Preprint |
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2021
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| _version_ | 1866909354668589056 |
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| author | Kang, Dong Yeap Kelly, Tom Kühn, Daniela Methuku, Abhishek Osthus, Deryk |
| author_facet | Kang, Dong Yeap Kelly, Tom Kühn, Daniela Methuku, Abhishek Osthus, Deryk |
| contents | In 1977, Erdős asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the largest possible chromatic number of the union $\bigcup_{i=1}^{n} G_i$? The equivalent dual formulation of this question asks for the largest chromatic index of an $n$-vertex hypergraph with maximum degree at most $n$ and maximum codegree at most $t$. For the case $t = 1$, Erdős, Faber, and Lovász famously conjectured that the answer is $n$, which was recently proved by the authors for all sufficiently large $n$. In this paper, we answer this question of Erdős for $t \geq 2$ in a strong sense, by proving that every $n$-vertex hypergraph with maximum degree at most $(1-o(1))tn$ and maximum codegree at most $t$ has chromatic index at most $tn$ for any $t,n \in \mathbb{N}$. Moreover, equality holds if and only if the hypergraph is a $t$-fold projective plane of order $k$, where $n = k^2 + k + 1$. Thus, for every $t \in \mathbb N$, this bound is best possible for infinitely many integers $n$. This result also holds for the list chromatic index. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_06181 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Solution to a problem of Erdős on the chromatic index of hypergraphs with bounded codegree Kang, Dong Yeap Kelly, Tom Kühn, Daniela Methuku, Abhishek Osthus, Deryk Combinatorics In 1977, Erdős asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the largest possible chromatic number of the union $\bigcup_{i=1}^{n} G_i$? The equivalent dual formulation of this question asks for the largest chromatic index of an $n$-vertex hypergraph with maximum degree at most $n$ and maximum codegree at most $t$. For the case $t = 1$, Erdős, Faber, and Lovász famously conjectured that the answer is $n$, which was recently proved by the authors for all sufficiently large $n$. In this paper, we answer this question of Erdős for $t \geq 2$ in a strong sense, by proving that every $n$-vertex hypergraph with maximum degree at most $(1-o(1))tn$ and maximum codegree at most $t$ has chromatic index at most $tn$ for any $t,n \in \mathbb{N}$. Moreover, equality holds if and only if the hypergraph is a $t$-fold projective plane of order $k$, where $n = k^2 + k + 1$. Thus, for every $t \in \mathbb N$, this bound is best possible for infinitely many integers $n$. This result also holds for the list chromatic index. |
| title | Solution to a problem of Erdős on the chromatic index of hypergraphs with bounded codegree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2110.06181 |