The saturation number of $K_{3,3}$

TL;DR

This paper proves the conjecture that the saturation number of the complete bipartite graph $K_{3,3}$ is exactly $3n-9$ for all sufficiently large $n$, advancing understanding of saturation numbers in graph theory.

Contribution

The paper confirms a conjecture that the saturation number of $K_{3,3}$ is $3n-9$ for all $n geq 9$, providing a key result in the study of saturation numbers.

Findings

Proves $sat(n, K_{3,3})=3n-9$ for $n geq 9$

Advances the understanding of saturation numbers for bipartite graphs

Completes a conjecture in the field of graph saturation

Abstract

A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. Determining the saturation number of complete partite graphs is one of the most important problems in the study of saturation number. The value of $sat(n,K_{2,2})$ was shown to be $\lfloor\frac{3n-5}{2}\rfloor$ by Ollmann, and a shorter proof was later given by Tuza. For $K_{2,3}$, there has been a series of study aiming to determine $sat(n,K_{2,3})$ over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that $sat(n, K_{2,3})= 2n-3$ for all $n\geq 5$. In this paper, we prove a conjecture of Pikhurko and Schmitt that $sat(n, K_{3,3})=3n-9$ when $n \geq 9$.