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1. If a graph has an Euler path, then it is planar.

2. If a graph does not have an Euler path, then it is not planar.

3. There is a graph which is planar and does not have an Euler path.

4. Yes. In fact, in this case it is because the original statement is false.

5. False. \(K_4\) is planar but does not have an Euler path.

6. False. \(K_5\) has an Euler path but is not planar.