Abstract
For fixed graphs $F$ and $H$, a graph $G\subseteq F$ is $H$-saturated if there is no copy of $H$ in $G$, but for any edge $e\in E(F)\setminus E(G)$, there is a copy of $H$ in $G+e$. The saturation number of $H$ in $F$, denoted $sat(F,H)$, is the minimum number of edges in an $H$-saturated subgraph of $F$. In this paper, we study saturation numbers of $tK_{l,l,l}$ in complete tripartite graph $K_{n_1,n_2,n_3}$. For $t\ge 1$, $l\ge 1$ and $n_1,n_2$ and $n_3$ sufficiently large, we determine $sat(K_{n_1,n_2,n_3},tK_{l,l,l})$ exactly.
Highlights
In this paper, we only consider finite, simple and undirected graphs
Hajnal and Moon [5] initiated the study of saturation numbers by determining sat(n, Kr) =
We begin our construction of a tKl,l,l-saturated graph, denoted by H, of Kn1,n2,n3
Summary
We only consider finite, simple and undirected graphs. Let G = (V, E) be a graph, where V is the vertex set and E is the edge set of G. We will use tH to denote t pairwise disjoint copies of H. Let Kn1,n2,n3 be a complete tripartite graph with ni vertices in the ith partite, where 1 i 3. A graph G is said to be H-saturated if it does not contain H as a subgraph, but the addition of any new edge from E(G) forms a copy of H, where G is the complement of G. Let sat(n, H) denote the minimal size of an H-saturated n-vertex graph. Hajnal and Moon [5] initiated the study of saturation numbers by determining sat(n, Kr) =.
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