Saturation Numbers in Tripartite Graphs

Abstract

Given graphs H and F, a subgraph G⊆H is an F‐saturated subgraph of H if F⊈G, but F⊆G+e for all e∈E(H)∖E(G). The saturation number of F in H, denoted sat (H,F), is the minimum number of edges in an F‐saturated subgraph of H. In this article, we study saturation numbers of tripartite graphs in tripartite graphs. For ℓ≥1 and n1, n2, and n3 sufficiently large, we determine sat (Kn1,n2,n3,Kℓ,ℓ,ℓ) and sat (Kn1,n2,n3,Kℓ,ℓ,ℓ−1) exactly and sat (Kn1,n2,n3,Kℓ,ℓ,ℓ−2) within an additive constant. We also include general constructions of Kℓ,m,p‐saturated subgraphs of Kn1,n2,n3 with few edges for ℓ≥m≥p>0.

Partite Saturation of Complete Graphs

“…They conjectured a value for the saturation number sat(K m,n , K r,r ) which was verified independently by Bollobás [2,3] and Wessel [15,16]. Very recently, Sullivan and Wenger [14] studied the analogous saturation numbers for tripartite graphs within tripartite graphs and determined sat(K n1,n2,n3 , K l,l,l ) for every fixed l ≥ 1 and every n 1 , n 2 and n 3 sufficiently large. Several other host graphs have been considered, including hypercubes [4,9,12] and random graphs [11].…”

Saturation Number of $tK_{l,l,l}$ in the Complete Tripartite Graph

“…Results about the saturation number when the host graphs are not complete can be foud in [1], [11]- [15]. In [16], Sullivan and Wenger studied saturation numbers in tripartite graphs and determined sat(K n 1 ,n 2 ,n 3 , K l,l,l ). In this paper, we generalize Sullivan and Wenger's result and determine sat(K n 1 ,n 2 ,n 3 , tK l,l,l ) exactly for t 1.…”

“…Before that, we need some lemmas. The idea of the proofs of the following two lemmas comes from [16]. Let…”