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Abstract
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 $\mathcal{G}(n,V_{r,3})$ denote the class of graphs on $n$ vertices containing no $r$-vertex-disjoint cycles of length $3$. We show that for large $n$, $\mathcal{E}(G)\les \lfloor \frac{(n-r+1)^2}{4} \rfloor +(r-1)(n-r+1)$ for every $G\in\mathcal{G}(n,V_{r,3})$. Furthermore, equality holds if and only if $G=\Omega(n,r)=K_{r-1,\lfloor \frac{n-r+1}2\rfloor,\lceil \frac{n-r+1}2\rceil}$ where $\Omega(n,r)$ is a tripartite graph on $n$ vertices.
Highlights
In this paper, we only consider simple graphs with vertex set V(G) and edge set E(G)
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
The cycle on n vertices is denoted by Cn and the complete tripartite graph with partitioning sets of order m, n and k is denoted by Km,n,k
Summary
We only consider simple graphs with vertex set V(G) and edge set E(G). 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. G ∈ G(n; V2k+1) where G(n; V2k+1) is the class of graphs on n vertices containing no vertex-disjoint cycles of length 2k + 1. M. M., 2012) in which we considered here no r vertex-disjoint cycles of length 3 instead of edge-disjoint cycles discussed in
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