Question

Prove that if the complete graph Kn can be decomposed into 5-cycles (i.e., each edge of Kn appears in exactly one of the 5-cycles of the decomposition), then n-1 or n-5 is divisiable by 10.

Answer #1

GRAPH THEORY:
Let G be a graph which can be decomposed into Hamilton
cycles.
Prove that G must be k-regular, and that k must be even.
Prove that if G has an even number of vertices, then the edge
chromatic number of G is Δ(G)=k.

A Hamiltonian cycle is a graph cycle (i.e., closed loop) through
a graph that visits each vertex exactly
once. A graph is called Hamiltonian if it contains a Hamiltonian
cycle. Suppose a graph is composed of
two components, both of which are Hamiltonian.
Find the minimum number of edges that one needs to add to obtain
a Hamiltonian graph. Prove your answer.

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.

Complete the following table accurately.
[5 Marks] Draw the TC,
MR, MC in one graph
Q
TFC
TVC
TC
P=MR
TR
MC
Profit
0
$10
0
$15
1
10
2
15
3
20
4
30
5
50
6
80

Complete the following table accurately.
[5 Marks] Draw the TFC,
AFC and AVC in one graph
Q
TVC
TFC
TC
MC
ATC
AFC
AVC
0
0
100
1
20
2
38
3
51
4
62
5
75
6
90

A graph is k-colorable if each vertex can be assigned one of k
colors so that no two adjacent vertices have the same color. Show
that a graph with maximum degree at most r is (r +
1)-colorable.

Suppose 5 distinct balls are distributed into 3 distinct boxes
such that each of the 5 balls can get into any of the 3 boxes.
1) What is the Probability that box 1 has exactly two balls and
the remaining balls are in the other two boxes.
2) What is the probability that there is exactly one empty
box?

For each of the statements below, say what method of proof you
should use to prove them. Then say how the proof starts and how it
ends. Pretend bonus points for filling in the middle.
a. There are no integers x and y such that x is a prime greater
than 5 and x = 6y + 3.
b. For all integers n , if n is a multiple of 3, then n can be
written as the sum of...

In the article “The Eastern Cottonmouth (Agkistrodon
piscivorus) at the Northern Edge of Its Range” (Journal of
Herpetology, Vol. 29, No. 3, pp. 391-398), C. Blem and L. Blem
examined the reproductive characteristics of the eastern
cottonmouth snake. The data, provided in the attached file, give
the number of young per litter for 24 female cottonmouths in
Florida and 44 female cottonmouths in Virginia. Preliminary data
analyses indicate that you can reasonably presume that the litter
sizes of cottonmouths in...

A ‘To Do’ list contains 5 tasks unrelated to each other. All
tasks have to be completedwithin 3 days. It is necessary to do at
least 1 task everyday. How many different ways are there to make a
schedule to complete these 5 tasks?
Generalize the result obtained in the previous exercise to n
tasks and k days (n ≥ k ; there is at least one task scheduled for
every day).

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