Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to Ks).
Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to Ks).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Ouestion 2
(a) Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to
Ks).
(b) Write down two induced subgraphs of the Petersen graph.
(c) Show that the simple connected planar graph with 17 edges and 10 vertices cannot be properly
coloured with two colours. [Hint: Show that such a graph must contain a triangle. Prove by
contradiction].
(d) Suppose G is a connected, simple finite planar graphs with n > k vertices. Show that G has at
most (n - 2) edges.
(e) Let G be a connected planar graph with 20 vertices each of degree 3. Find the number of
regions in the graph.
(f) Prove that a planar graph with n 2 4 vertices in which every vertex has degree at least 2 has at
least 4 vertices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdde0a7f2-24be-48bf-a697-b631f9ae73a3%2Ff5a77343-d707-49fb-9cfd-f72bf0cb81a0%2F0o6t80s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ouestion 2
(a) Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to
Ks).
(b) Write down two induced subgraphs of the Petersen graph.
(c) Show that the simple connected planar graph with 17 edges and 10 vertices cannot be properly
coloured with two colours. [Hint: Show that such a graph must contain a triangle. Prove by
contradiction].
(d) Suppose G is a connected, simple finite planar graphs with n > k vertices. Show that G has at
most (n - 2) edges.
(e) Let G be a connected planar graph with 20 vertices each of degree 3. Find the number of
regions in the graph.
(f) Prove that a planar graph with n 2 4 vertices in which every vertex has degree at least 2 has at
least 4 vertices.
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