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+10 votes

Which of the following statements is true for every planar graph on $n$ vertices?

  1. The graph is connected
  2. The graph is Eulerian
  3. The graph has a vertex-cover of size at most $\frac{3n}{4}$
  4. The graph has an independent set of size at least $\frac{n}{3}$
asked in Graph Theory by Veteran (68.8k points)
retagged by | 1.3k views
This is not out of syllabus.
not out of syllabus. 2016 question came from here

3 Answers

+10 votes
Best answer

Planar graph not  necesarily be connected. So option (A) is false. Options (B) and (D) can be proven false with the example of K4 . Now only one option (C)  left  .

So, answer should be (C). But I don't know how to prove it.

answered by Boss (5.6k points)
selected by
How is option D proved false with example of K4? K4 has an independent set of size 1 which is what you get by doing n/3 => 4/3 = 1.
For vertex cover of K4 size will be 2 maximal two edges would be sufficient to cover all vertices here and according to formula it would be less than or equal to (3*4)/4 equals to 3 bt how size of vertex cover would be 3 not getting it 2 becoz in that case it wouldn't be maximal set
We can prove C using every planar graph is 4 colourable.
A wheel with 7 vertices is counter example for option D.

The independent set can consist of centre vertex which alone forms an independent set, whose size is 1. (given size is atleast n/3, here 7/3 = 2).So it is wrong.

Is this right?
+9 votes

Independent Set ≥ ceil(n/k) where,
n is the number of vertices and k is the chromatic number 
For any planar graph k ≥ 3 and k ≤ 4, therefore, the Independent set is at least ceil(n/4).
Hence (D) is false.

Now we know that Vertex cover + Independent Set Number =n.
If Independent set ≥ ceil(n/4) then,
Vertex cover ≤ n - (n/4)
Vertex Cover ≤ 3n/4

Vertex Cover is at most 3n/4. So (C) is the correct answer.


answered by (225 points)
I can't find any source for the formula for independent set$\geq \left \lceil n/k \right \rceil$. Please cite your source.
@habedo, we can use our Intuition here.IF we can color a graph using $k$ colors then it means every set of $k$ vertices are dependent (or adjacent ) to each other so  total size of independent set  will be $n/k$
–2 votes
Answer is (A) The graph is connected
answered by Boss (6.3k points)
Yeah it should be C. I came to the same conclusion, following the same reasoning
Planar graph dont need to be connected.

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