Planar Graph

Question

Can minimum degree of a planar graph be $5$? Give some example

Comments

Already mentioned in above comment 

a graph may follow E<=3v-6 but it does not mean it is planar.

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@Satbir ur 2nd graph is wrong. Plz remove it. It has many vertex with degree 3. Only @Gyanu graph is correct. 

More over  graph neednot to be complete. But yes degree 5 graph is possible and degree 6 graph not possible

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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.

Answer 1

For a complete planar graph Minimum degree of the graph could not be $5$. It could be $3$ at max.

However, the degree of one of the vertex in planar graph can be $5$

Also we can have a regular planar graph with degree $5$ at max.

Comments

@Satbir if we do like this, then a planar graph with degree 6 also possible. rt??

But according to them(GATE-2003 question) planar graph with degree 5 possible, but planar graph with degree 6 not possible.

Here my question was , why??

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Yes, if we place a vertex in the middle and add connect it to other vertex such that edges do not intersect then we can have degree 7 also. similarly we can have any number of degree for a vertex.

Please note that they are asking for min degree of G in gate 2003.