Saturation numbers in tripartite graphs

TL;DR

This paper investigates the minimum number of edges in tripartite graphs that are saturated with respect to certain complete tripartite subgraphs, providing exact and near-exact formulas for large vertex classes.

Contribution

It determines exact saturation numbers for specific tripartite subgraphs in large tripartite graphs and offers general constructions for such saturated subgraphs.

Findings

Exact saturation numbers for $K_{ ext{ell}, ext{ell}, ext{ell}}$ and $K_{ ext{ell}, ext{ell}, ext{ell}-1}$.

Approximate saturation numbers for $K_{ ext{ell}, ext{ell}, ext{ell}-2}$.

Constructive methods for $K_{ ext{ell},m,p}$-saturated subgraphs with few edges.

Abstract

Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F\nsubseteq G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$. The saturation number of $F$ in $H$, denoted $\text{sat}(H,F)$, is the minimum number of edges in an $F$-saturated subgraph of $H$. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For $\ell\ge 1$ and $n_1$, $n_2$, and $n_3$ sufficiently large, we determine $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell})$ and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-1})$ exactly and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-2})$ within an additive constant. We also include general constructions of $K_{\ell,m,p}$-saturated subgraphs of $K_{n_1,n_2,n_3}$ with few edges for $\ell\ge m\ge p>0$.