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

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.

The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

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.

Power of 2 Decomposition of a Complete Tripartite Graph K2,4,M and a Special Butterfly Graph

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.

Packing Graphs of Bounded Codegree

Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that if (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumption that G1 or G2 has bounded codegree. In particular, we prove for all t ⩾ 2 that if G1 contains no copy of the complete bipartite graph K2,t and Δ(G1) > 17t · Δ(G2), then (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1 implies that G1 and G2 pack. We also provide a mild improvement if moreover G2 contains no copy of the complete tripartite graph K1,1,s, s ⩾ 1.