Question

Prove the number of vertices of degree 1 in an tree must be
greater than or equal to the maximum degree in the tree.

(Try either Contradiction or Direct Proof)

Answer #1

please use contradiction
Prove the number of vertices of degree 1 in a tree must be
greater than or equal to the maximum degree in the tree.

(1) Let x be a rational number and y be an irrational. Prove
that 2(y-x) is irrational
a) Briefly explain which proof method may be most appropriate to
prove this statement. For example either contradiction,
contraposition or direct proof
b) State how to start the proof and then complete the proof

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

In lecture, we proved that any tree with n vertices must have n
− 1 edges. Here, you will prove the converse of this statement.
Prove that if G = (V, E) is a connected graph such that |E| =
|V| − 1, then G is a tree.

prove that if u and v are distinct vertices in a rooted tree
which share a common descendant, then either u is the descendant of
v or v is the descendant of u

Graph Theory
.
While it has been proved that any tree with n vertices must have
n − 1 edges. Here, you will prove the converse of this statement.
Prove that if G = (V, E) is a connected graph such that |E| = |V |
− 1, then G is a tree.

Prove by contradiction that:
If n is an integer greater than 2, then for all integers m, n
does not
divide m or n + m ≠ nm.

Let G be a simple planar graph with fewer than 12
vertices.
a) Prove that m <=3n-6; b) Prove that G has a vertex of degree
<=4.
Solution: (a) simple --> bdy >=3. So 3m - 3n + 6 = 3f
<= sum(bdy) = 2m --> m - 3n + 6 <=0 --> m <= 3n -
6.
So for part a, how to get bdy >=3 and 2m? I need a
detailed explanation
b) Assume all deg >= 5...

Prove that a natural number m greater than 1 is prime if m has
the property that it divides at least one of a and b whenever it
divides ab.

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

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