NIELIT 2017 July Scientist B (CS) - Section B: 14

Question

If $G$ is an undirected planar graph on $n$ vertices with $e$ edges then

• A. $e\leq n$
• B. $e\leq 2n$
• C. $e\leq 3n$
• D. None of the option

Answer 1

for planar graphs, 3r <= 2e and v - e + r = 2

From these we get  e <= 3v - 6

Now 2v <= 3v - 6 for all vertices v >=3

So e <= 2n is the answer.

Answer 2

Theorem 1 (Euler's Formula)   Let $G$ be a connected planar graph, and let $n, m$ and $f$ denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of $G.$ Then $n - m + f = 2.$

Corollary 1    Let $G$ be a connected planar simple graph with $n$ vertices, where $n \geq 3$ and $m$ edges. Then $m \leq 3n - 6.$

Corollary 2     Let $G$ be a connected planar simple graph with $n$ vertices and $m$ edges, and no triangles. Then $m \leq 2n - 4.$

Reference 

Answer 3

Option B

The Handshaking lemma

see page 5 https://www.cs.cmu.edu/~adamchik/21-127/lectures/graphs_1_print.pdf

Answer 4

A planar graph of order n and size m. Then, m ≤ 3n-6.