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

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

Let $G$ be a graph. We say an $r$-uniform hypergraph $H$ is a Berge-$G$ if there exists a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each $e\in E(G)$. Given a family of $r$-uniform hypergraphs $\mathcal{F}$ and an $r$-uniform hypergraph $H$, a spanning sub-hypergraph $H'$ of $H$ is $\mathcal{F}$-saturated in $H$ if $H'$ is $\mathcal{F}$-free, but adding any edge in $E(H)\backslash E(H')$ to $H'$ creates a copy of some $F\in\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges in an $\mathcal{F}$-saturated spanning sub-hypergraph of $H$. In this paper, we asymptotically determine the saturation number of Berge stars in random $r$-uniform hypergraphs.

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number $\operatorname{sat}(\mathcal{F},n)$ is the minimum number of edges in an $\mathcal{F}$-saturated graph on $n$ vertices. We prove that there exists a finite family $\mathcal{F}$ such that $\operatorname{sat}(\mathcal{F},n) / n^{r-1}$  does not tend to a limit. This settles a question of Pikhurko.

For a fixed graph $H$, a graph $G$ is $H$-saturated if there is no copy of $H$ in $G$, but for any edge $e \notin G$, there is a copy of $H$ in $G + e$. The collection of $H$-saturated graphs of order $n$ is denoted by ${\bf SAT}(n,H)$, and the saturation number, ${\bf sat}(n, H),$ is the minimum number of edges in a graph in ${\bf SAT}(n,H)$. Let $T_k$ be a tree on $k$ vertices. The saturation numbers ${\bf sat}(n,T_k)$ for some families of trees will be determined precisely. Some classes of trees for which ${\bf sat}(n, T_k) < n$ will be identified, and trees $T_k$ in which graphs in ${\bf SAT}(n,T_k)$ are forests will be presented. Also, families of trees for which ${\bf sat}(n,T_k) \geq n$ will be presented. The maximum and minimum values of ${\bf sat}(n,T_k)$ for the class of all trees will be given. Some properties of ${\bf sat}(n,T_k)$ and ${\bf SAT} (n,T_k)$ for trees will be discussed.

Let G be a finite, connected simple graph with p vertices and q edges. If G1 , G2 ,…, Gn are connected edge-disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  …  E(Gn) , then {G1 , G2 , …, Gn} is said to be a decomposition of G. A graph G is said to have Power of 2 Decomposition if G can be decomposed into edge-disjoint subgraphs G G G n  2 4 2 , ,..., such that each G i 2 is connected and ( ) 2 , i E Gi  for 1  i  n. Clearly, 2[2 1] n q . In this paper, we investigate the necessary and sufficient condition for a complete tripartite graph K2,4,m and a Special Butterfly graph           3 2 5 BF 2m 1 to accept Power of 2 Decomposition.