Density conditions for $k$ vertex-disjoint triangles in tripartite graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929746253709312 |
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| author | Guo, Mingyang Markström, Klas |
| author_facet | Guo, Mingyang Markström, Klas |
| contents | Let $n,k$ be positive integers such that $n\geq k$ and $G$ be a tripartite graph with parts $A,B,C$ such that $|A|=|B|=|C|=n$. Denote the edge densities of $G[A,B]$, $G[A,C]$ and $G[B,C]$ by $α$, $β$ and $γ$, respectively. In this paper, we study edge density conditions for the existence of $k$ vertex-disjoint triangles in a tripartite graph. For $n\geq 5k+2$ we give an optimal condition in terms of densities $α,β,γ$ for the existence of $k$ vertex-disjoint triangles in $G$. We also give an optimal condition in terms of densities $α,β,γ$ for the existence of a triangle-factor in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density conditions for $k$ vertex-disjoint triangles in tripartite graphs Guo, Mingyang Markström, Klas Combinatorics Let $n,k$ be positive integers such that $n\geq k$ and $G$ be a tripartite graph with parts $A,B,C$ such that $|A|=|B|=|C|=n$. Denote the edge densities of $G[A,B]$, $G[A,C]$ and $G[B,C]$ by $α$, $β$ and $γ$, respectively. In this paper, we study edge density conditions for the existence of $k$ vertex-disjoint triangles in a tripartite graph. For $n\geq 5k+2$ we give an optimal condition in terms of densities $α,β,γ$ for the existence of $k$ vertex-disjoint triangles in $G$. We also give an optimal condition in terms of densities $α,β,γ$ for the existence of a triangle-factor in $G$. |
| title | Density conditions for $k$ vertex-disjoint triangles in tripartite graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.05218 |