Question

Let G be a simple graph in which all vertices have degree four.
Prove that it is possible to color the edges of G orange or blue so
that each vertex is adjacent to two orange edges and two blue
edges.

Hint: The graph G has a closed Eulerian walk. Walk along it and
color the edges alternately orange and blue.

Answer #1

Suppose we are going to color the vertices of a connected planar
simple graph such that no two adjacent vertices are with the same
color.
(a) Prove that if G is a connected planar simple graph, then G
has a vertex of degree at most five.
(b) Prove that every connected planar simple graph can be
colored using six or fewer colors.

I.15: If G is a simple graph with at least two vertices, prove
that G has two vertices of the same degree.
Hint: Let G have n vertices. What are possible
different degree values? Different values if G is connected?

Prove or disapprove each of the following:
(a) Every disconnected graph has an isolated vertex.
(b) A graph is connected if and only if some vertex is connected
to all other vertices.
(c) If G is a simple, connected, Eulerian graph, with edges e, f
that are incident to a common vertex, then G has an Eulerian
circuit in which e and f appear consequently.

Graph Theory.
A simple graph G with 7 vertices and 10 edges has the
following properties: G has six vertices of degree
a and one vertex of degree b. Find a and
b, and draw the graph.
Show all work.

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...

Let
G be a simple graph with at least two vertices. Prove that there
are two distinct vertices x, y of G such that deg(x)= deg(y).

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 a simple graph with p vertices and q edges is complete
(has all possible edges) if and only if q=p(p-1)/2.
please prove it step by step. thanks

Question 38
A simple connected graph with 7 vertices has 3 vertices of
degree 1, 3 vertices of degree 2 and 1 vertex of degree 3. How many
edges does the graph have?
Question 29
Use two of the following sets for each part below. Let X = {a,
b, c}, Y = {1, 2, 3, 4} and Z = {s, t}. a) Using ordered pairs
define a function that is one-to-one but not onto. b) Using ordered
pairs define...

Let u and v be distinct vertices in a graph G. Prove that there
is a walk from ? to ? if and only if there is a path from ? to
?.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 6 minutes ago

asked 20 minutes ago

asked 26 minutes ago

asked 31 minutes ago

asked 36 minutes ago

asked 37 minutes ago

asked 43 minutes ago

asked 43 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago