$n$ vertices be denoted as $K_n$ .

Suppose there are 2015 vertices. Let a complete graph onWhich statements are true?

I. $K_{2015}$ has 2029105 edges.

II. Suppose two $K_{2015}$ graphs are connected together, by having exactly one edge connected between any vertex in each of the original $K_{2015}$ graphs, to form a new graph $G_{2015}$ . Then $G_{2015}$ is bipartite.

III. Let $H_{2014}$ be a complete 2014-partite graph on $2014^2$ vertices, such that the number of vertices in each partition are equal. Then $K_{2015}$ is not a subgraph of $H_{2014}$ .

*
This problem is part of the set
2015 Countdown Problems
.
*

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