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arXiv · 2503.05218

Density conditions for $k$ vertex-disjoint triangles in tripartite graphs

Mingyang Guo·Klas Markström
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Abstract

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 $\alpha$, $\beta$ and $\gamma$, 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 $\alpha,\beta,\gamma$ for the existence of $k$ vertex-disjoint triangles in $G$. We also give an optimal condition in terms of densities $\alpha,\beta,\gamma$ for the existence of a triangle-factor in $G$.

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Mingyang Guo, Klas Markström. 2025-03-07. Density conditions for $k$ vertex-disjoint triangles in tripartite graphs. https://arxiv.org/abs/2503.05218

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