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.

Let ? be a connected graph with at least one edge.
(a) Prove that each vertex of ? is saturated by
some maximum matching in ?.
(b) Prove or disprove the following: Every edge
of ? is in some maximum matching of ?.

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 if G is a connected graph with exactly 4 vertices of
odd degree, there exist two trails in G such that each edge is in
exactly one trail. Find a graph with 4 vertices of odd degree
that’s not connected for which this isn’t true.

(a) Prove that there does not exist a graph with 5 vertices with
degree equal to 4,4,4,4,2.
(b) Prove that there exists a graph with 2n vertices with
degrees 1,1,2,2,3,3,..., n-1,n-1,n,n.

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.

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

Prove that for each k ≥ 1, a graph which is regular
with degree 2k can never have a bridge.

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

Suppose G is a simple, nonconnected graph with n vertices that
is maximal with respect to these properties. That is, if you tried
to make a larger graph in which G is a subgraph, this larger graph
will lose at least one of the properties (a) simple, (b)
nonconnected, or (c) has n vertices.
What does being maximal with respect to these properties imply
about G?G? That is, what further properties must GG possess because
of this assumption?
In this...

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