{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Discrete Mathematics with Graph Theory (3rd Edition) 365

# Discrete Mathematics with Graph Theory (3rd Edition) 365 -...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Section 13.1 363 we have V3 = V2 + 1, while E3 2: E2 + 1. Hence, E2 - V2 ~ E3 - V3 and so EI - VI < E3 - V3. Thus, 9 and its sub graph K provide another counterexample, but this contradicts the minimality of VI - V2 since VI - V3 < VI - V2. (b) Say 9 is not planar. Then 9 has a subgraph 1i with VI vertices and EI edges which is homeomor- phic to K3 ,3 or K 5 • For K3,3, E - V = 9 - 6 = 3. For K 5 , E - V = 10 - 5 = 5. By Exercises 12 and 13, we conclude that EI - Vi = 3 or 5, but we are given that EI - VI ~ 2, contradicting part (a). (c) The graph need not be planar if E = V + 3 since K 3 ,3 has six vertices and nine edges. 15. [BB] Yes. An example is shown to the right. The graphs are homeomorphic since the one on the right is obtainable from the other by adding a vertex of degree 2. G----E) 0 0 0 A B A B 16. Doubtlessly Poland's most famous mathematician, Kazimierz Kuratowski was born in 1896 in Warsaw where he died on the 18th of June, 1980. Renowned for his lectures, Kuratowski's research was where he died on the 18th of June, 1980....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online