The saturation number for unions of four cliques

TL;DR

This paper determines the minimal number of edges in graphs that are saturated with respect to unions of four cliques under specific size conditions, extending previous work on unions of three cliques.

Contribution

It extends the known results on saturation numbers to unions of four cliques with certain size constraints, providing exact values for these saturation numbers.

Findings

Derived the saturation number for unions of four cliques with specified size differences.

Extended previous results from three to four clique unions.

Provided exact formulas for the minimal edges in such saturated graphs.

Abstract

A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. The $sat(n, K_{p_1} \cup K_{p_2} \cup K_{p_3})$ with $p_3 \ge p_1 + p_2$ was given in [Discrete Math. 347 (2024) 113868]. In this paper, $sat(n,K_{p_1} \cup K_{p_2} \cup K_{p_3} \cup K_{p_4})$ with $p_{i+1} - p_i \ge p_1$ for $2 \le i\le 3$ and $4\le p_1\le p_2$ is determined.