On multicolor Turán numbers
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910494892228608 |
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| author | Balogh, József Liebenau, Anita Mattos, Letícia Morrison, Natasha |
| author_facet | Balogh, József Liebenau, Anita Mattos, Letícia Morrison, Natasha |
| contents | We address a problem which is a generalization of Turán-type problems recently introduced by Imolay, Karl, Nagy and Váli. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$, where each $F_i$ is isomorphic to a fixed graph $F$ and $E(F_i)\cap E(F_j)=\emptyset$ for all $i \neq j$. We call a subgraph $H\subseteq G$ multicolored if $H$ and $F_i$ share at most one edge for all $i$. Define $\text{ex}_F(H,n)$ to be the maximum value $k$ such that there exists $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$ on $n$ vertices without a multicolored copy of $H$. We show that $\text{ex}_{C_5}(C_3,n) \le n^2/25 + 3n/25+o(n)$ and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On multicolor Turán numbers Balogh, József Liebenau, Anita Mattos, Letícia Morrison, Natasha Combinatorics We address a problem which is a generalization of Turán-type problems recently introduced by Imolay, Karl, Nagy and Váli. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$, where each $F_i$ is isomorphic to a fixed graph $F$ and $E(F_i)\cap E(F_j)=\emptyset$ for all $i \neq j$. We call a subgraph $H\subseteq G$ multicolored if $H$ and $F_i$ share at most one edge for all $i$. Define $\text{ex}_F(H,n)$ to be the maximum value $k$ such that there exists $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$ on $n$ vertices without a multicolored copy of $H$. We show that $\text{ex}_{C_5}(C_3,n) \le n^2/25 + 3n/25+o(n)$ and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term. |
| title | On multicolor Turán numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.05060 |