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

Abstract

Let $n,k$ be positive integers such that $n\ge k$ 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 $n\ge 5k+2$ 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 $G$.