Edge-Maximal Graphs Containing No r Vertex-Disjoint Triangles

An important problem in graph theory is that of determining the maximum number of edges in a given graph G that contains no specific subgraphs. This problem has attracted the attention of many researchers. An example of such a problem is the determination of an upper bound on the number of edges of a graph that has no triangles. In this paper, we let G(n,Vr,3) denote the class of graphs on n vertices containing no r-vertex-disjoint cycles of length 3. We show that for large n, E(G) 6 ⌊ (n−r+1)2 4 ⌋ + (r − 1)(n − r + 1) for every G ∈ G(n,Vr,3). Furthermore, equality holds if and only if G = Ω(n, r) = Kr−1,⌊ n−r+1 2 ⌋,⌈ n−r+1 2 ⌉ where Ω(n, r) is a tripartite graph on n vertices.


Introduction
In this paper, we only consider simple graphs with vertex set V(G) and edge set E(G).If an edge e ∈ E(G) is incident with the two vertices u and v in V(G), we write e = uv = vu.For a vertex u ∈ V(G) the neighborhood of u, denoted by N G (u), is the set of vertices v ∈ V(G) such that uv ∈ E(G).The degree d G (u) is the cardinality of N G (u).
For vertex-disjoint subgraphs H 1 and H 2 of G, we let E(H 1 , H 2 ) to be the set of all edges that are incident to a vertex in H 1 and a vertex in H 2 .That is E(H 1 , H 2 ) = {uv ∈ E(G) | u ∈ V(H 1 ), v ∈ H 2 }.We also define E(G) to be the cardinality of E(G) and E(H 1 , H 2 ) = |E(H 1 , H 2 )|.The cycle on n vertices is denoted by C n and the complete tripartite graph with partitioning sets of order m, n and k is denoted by K m,n,k .For given graphs G 1 and G 2 we denote the union of G 1 and G 2 by An important problem in extremal graph theory is the determination of maximum number of edges a graph has under a condition that the given graph has no specific subgraphs.Such an example is finding an upper bound for E(G) whenever G has no triangles (cycles of length 3) or, in general, G has no odd disjoint cycles.We have two types of disjoint cycles, the first type is edge-disjoint cycles, and the second type is vertex-disjoint cycles.Note that vertex-disjoint cycles are edge-disjoint cycles, but not vice-versa.
The determination of maximum number of edges in a graph that forbids certain subgraphs has attracted the attention of many graph researchers.For example, Höggkvist et al in (Höggkvist, R., Faudree, R. J., & Schelp, R. H., 1981) proved that E(G) ⌊ (n−1) 2 4 ⌋ + 1 for a non bipartite graph G with n vertices that contains no odd cycle C 2k+1 for all positive integers k.In (Bataineh, M., & Jaradat, M. M. M. , 2012), M. Bataineh and M. Jaradat proved that E(G) ⌊ n 2 4 ⌋ + r − 1 for any graph G ∈ G(n; r, 2k + 1) for large n and r 2, k 1, where G(n; r, 2k + 1) is the set of all graphs on n vertices containing no r edge-disjoint cycles of length 2k + 1.In (Bataineh, M.) is the class of graphs on n vertices containing no vertex-disjoint cycles of length 2k + 1.
In this paper, we will generalize a result that is parallel to the result of (Bataineh, M., & Jaradat, M. M. M., 2012) in which we considered here no r vertex-disjoint cycles of length 3 instead of edge-disjoint cycles discussed in (Bataineh, M., & Jaradat, M. M. M., 2012).

Important Lemmas and Theorems
In this section, we introduced necessary background that are needed in proving the main results of this paper.

Main Result
In this section, we generalize a special case of Theorem 2.3 to the case where G ∈ G(n, V r , 3).That is to the case where G is a graph on n vertices containing no r vertex-disjoint cycles of length 3. We start with r = 2.
Proof.Since G ∈ G(n, 2, 2k + 1), then G has no two vertex-disjoint cycles of length 3. Suppose first that G has no cycle of length 3. Then for n 11, we have 3 ⌊ n+3 3 ⌋, so that, using Lemma 2.1 we have: Therefore, equality holds if and only if G = Ω(n, 2).
To prove the main theorem we have to introduce Turán graphs since these graphs play a major role in the proof.
3.2 Definition.The complete s-partite graph on n vertices with part sizes being ⌋ is called Turán graph.We denote this graph by T n,s .
Note that Turán graph is K s+1 free, where K s+1 is the complete graph on (s + 1)-vertices.In (Conlon, D.), David Conlon introduced the following statement of Turán's theorem.

Theorem. (Turán) If G is an n-vertex K s+1 -free graph, then it contains at most E(T n,s ) edges.
In addition, Conlon introduced three different proofs of Turáns Theorem.In proof 2 (Zykovs Symmetrization), he concluded that the set of vertices of a K s+1 -free graph G on n vertices with maximum number of edges can be partitioned into two equivalence classes.In these classes, vertices in the same class are non-adjacent and vertices in different classes are adjacent.Since the graph G is K s+1 -free, it must be a complete s-partite graph.Note that T n,s is the unique graph that maximizes the number of edges among such graphs.

. ,C r−1 , but no r vertex disjoint cycles of length 3 and let H
Proof.Note that H is K 3 free graph since, otherwise, G would have r vertex-disjoint cycles of length 3, a contradiction to the assumption.Let H ′ be a graph on the vertices of H with a maximum number of edges.Note that that vertices in H ′ 1 are non-adjacent and, also, vertices in H ′ 2 are non-adjacent, but vertices in H ′ 1 are adjacent to vertices in H ′ 2 .In Figure 1, let 1 then we will have r vertex-disjoint cycles of length 3, a contradiction.It follows that w 1 and u 1 are adjacent to H ′ 1 or H ′ 2 but not to both.Therefore, 2(n − r + 1) − 4(r − 1) 2 .