Question

9.

Prove Euler's formula using induction on the number of
*vertices* in the graph.

Answer #1

Prove that the order of complete graph on n ≥ 2 vertices is
(n−1)n 2 by...
a) Using theorem Ʃv∈V = d(v) = 2|E|.
b) Using induction on the number of vertices, n for n ≥
2.

Use proof by induction to prove that every connected planar
graph with less than 12 vertices has a vertex of degree at most
4.

please solve it step by step. thanks
Prove that every connected graph with n vertices has at least
n-1 edges. (HINT: use induction on the number of vertices
n)

State the Division Algorithm for Natural number and prove it
using induction

Prove using induction: There is no rational number r
for which r2=2.

Draw an Euler graph on an even number of vertices, that does not
have perfect matching. Prove it.

Graph Theory, discrete math question:
Let G be a graph with 100 vertices, and chromatic number 99.
Prove a lower bound for the clique number of G. Any lower bound
will do, but try to make it as large as you can.
Please follow this hint my professor gave and show your work,
Thank you!!
Hint: can you prove that the clique number is at least 1? Now
how about 2? Can you prove that the clique number must be...

Show that the total degree of a complete graph with n nodes is
n(n-1) using INDUCTION.
Do not apply (a) the result on the total degree of a graph proven
(b) the formula for the number of edges in a complete graph.

Discrete math, graph theory question:
Let G be a graph with 100 vertices, and chromatic number 99.
Prove a lower bound for the clique number of G. (Hint: Any lower
bound will do, but try to make it as large as you can.)

Using induction, prove the following:
i.) If a > -1 and n is a natural number, then (1 + a)^n >=
1 + na
ii.) If a and b are natural numbers, then a + b and ab are also
natural

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