Density conditions for $k$ vertex-disjoint triangles in tripartite graphs
Mingyang Guo, Klas Markström
TL;DR
This work determines sharp density thresholds for packing $k$ vertex-disjoint triangles in a balanced tripartite graph with parts of size $n$. It introduces a $(k,n)$-cyclic triple condition on the three bipartite densities $α,β,γ$ and proves that, for $n\ge 5k+2$, this condition guarantees a $k$-triangle packing; a corresponding condition yields a triangle-factor. The approach combines inductive proofs with vertex-deletion lemmas that preserve cyclic-density inequalities, plus structured triad decomposition to bound remaining configurations. The results extend classical triangle-density questions to the tripartite setting and provide exact, best-possible density criteria for both disjoint triangles and triangle-factors, with discussion of asymptotic refinements and related hypergraph-matching bounds.
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 $α$, $β$ 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 $G$.