The saturation number of K3,3

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 bipartite graphs is one of the most important problems in the study of saturation numbers. The value of sat(n,K2,2) was shown to be ⌊3n−52⌋ by Ollmann, and a shorter proof was later given by Tuza. For K2,3, there has been a series of study aiming to determine sat(n,K2,3) over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that sat(n,K2,3)=2n−3 for all n≥5. Pikhurko and Schmitt conjectured that sat(n,K3,3)=(3+o(1))n. In this paper, for n≥9, we give an upper bound of 3n−9 for sat(n,K3,3), and prove that 3n−9 is also a lower bound when the minimum degree of a K3,3-saturated graph is 2 or 5, where it is trivial when the minimum degree is greater than 5.