Saturation Numbers in Tripartite Graphs

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 paper we study saturation numbers of tripartite graphs in tripartite graphs. For ` ≥ 1 and n1, n2, and n3 sufficiently large, we determine sat(Kn1,n2,n3 ,K`,`,`) and sat(Kn1,n2,n3 ,K`,`,`−1) exactly and sat(Kn1,n2,n3 ,K`,`,`−2) within an additive constant. We also include general constructions of K`,m,p-saturated subgraphs of Kn1,n2,n3 with few edges for ` ≥ m ≥ p > 0.