please note what you have asked in question
Can minimum degree of a planar graph be 5? Give some example
I have tried to provide various cases using examples just to clear confusion.
First graph is a complete graph and planar graph and min degree is 3... complete graph with degree >3 is not possible.
Second graph is simple graph and planar graph this i have used to show that degree in planar graph (not degree of planar graph ) can be >=5 just to avoid confusion regarding the statement which you are saying
degree 5 graph is possible and degree 6 graph not possible
graph with min degree 5 is possible and min degree 6 is not possible.your statement is also correct.
I have used the example just to show that degree of graph and degree of vertex are different things
Third graph is regular graph and planar graph and has min degree is 5. This is the main answer.
All the three satisfy E<=3v-6 since they are planar graph
If you still think it is not required then i will remove it.