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what is the matching number of $K_{2,3}$ graph.and also explain matching number of $K_{m,n}$(simplification).
asked in Graph Theory by Veteran (11.6k points) 6 20 86 | 64 views

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Matching number of a graph is defined as the maximum number of non-adjacent edges in a graph.

(I have marked one maximum-possible set of non-adjacent edges, but other set is also possible)

In a complete bipartite graph $K_{m,n}$ $m$ nodes are adjacent to each of the $n$ nodes and $n$ nodes are adjacent to each of $m$ nodes. (basic definition of bipartite graph)

So, If an edge is to be included in counting of Matching number, then all adjacent edges cannot be included. 

So, Matching Number in $K_{2,3} = 2$ and in $K_{3,3} = 3$

Although there may exist a much formal proof, but by taking few more examples it can be said that Matching Number for $\color{maroon}{K_{m,n} = \;min(m,n)}$

answered by Veteran (27.3k points) 19 70 243
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