# Discete Math

1) Let G be the graph whose vertices are all permutations of the integers of the set {1,2,?,10} with the property that two vertices (?1,?2,?,?10) , (??1,??2, ...,??10) are connected by an edge if one vertex arises from the other by exchanging the positions of two integers ?i,?j with ?i,?j?5

For example the vertices (1,**2**,7,4,8,**5**,3,10,9,6), (1,**5**,7,4,8,**2**,3,10,9,6) form an edge, vertices (1,**2**,4,8,**10**,7,3,5,9,6), (1,**10**,4,8,**2**,7,3,5,9,6) do not form an edge, vertices (1,**7**,2,4,8,**10**,3,5,9,6) , (1,**10**,2,4,8,**7**,3,5,9,6) do not form an edge. Find the number of edges of the graph G

2)How many induced subgraphs with 4 vertices does the complete graph K10 have? (We consider the vertices of K10 distinguished, i.e. named from 1 to 10)

3) The graph in the figure below is bipartite. Right or wrong?

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