On the multicolor Turán conjecture for color-critical graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866908541559767040 |
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| author | Li, Xihe Ma, Jie Zheng, Zhiheng |
| author_facet | Li, Xihe Ma, Jie Zheng, Zhiheng |
| contents | A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges receive distinct colors in a given simple $k$-coloring of $G$. In 2004, Keevash-Saks-Sudakov-Verstraëte introduced the {\it $k$-color Turán number} $ex_k(n,H)$, which denotes the maximum number of edges in an $n$-vertex multigraph that has a simple $k$-coloring containing no multicolored copies of $H$. They made a conjecture for any $r\geq 3$ and $r$-color-critical graph $H$ that in the range of $k\geq \frac{r-1}{r-2}(e(H)-1)$, if $n$ is sufficiently large, then $ex_k(n, H)$ is achieved by the multigraph consisting of $k$ colors all of which are identical copies of the Turán graph $T_{r-1}(n)$. In this paper, we show that this holds in the range of $k\geq 2\frac{r-1}{r}(e(H)-1)$, significantly improving earlier results. Our proof combines the stability argument of Chakraborti-Kim-Lee-Liu-Seo with a novel graph packing technique for embedding multigraphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_14905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the multicolor Turán conjecture for color-critical graphs Li, Xihe Ma, Jie Zheng, Zhiheng Combinatorics A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges receive distinct colors in a given simple $k$-coloring of $G$. In 2004, Keevash-Saks-Sudakov-Verstraëte introduced the {\it $k$-color Turán number} $ex_k(n,H)$, which denotes the maximum number of edges in an $n$-vertex multigraph that has a simple $k$-coloring containing no multicolored copies of $H$. They made a conjecture for any $r\geq 3$ and $r$-color-critical graph $H$ that in the range of $k\geq \frac{r-1}{r-2}(e(H)-1)$, if $n$ is sufficiently large, then $ex_k(n, H)$ is achieved by the multigraph consisting of $k$ colors all of which are identical copies of the Turán graph $T_{r-1}(n)$. In this paper, we show that this holds in the range of $k\geq 2\frac{r-1}{r}(e(H)-1)$, significantly improving earlier results. Our proof combines the stability argument of Chakraborti-Kim-Lee-Liu-Seo with a novel graph packing technique for embedding multigraphs. |
| title | On the multicolor Turán conjecture for color-critical graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.14905 |