Question

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.

Answer #1

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.

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...

Draw an example of a connected bipartite simple graph with 9
vertices and 10 edges that has an Euler tour.

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.

A K-regular graph G is a graph such that deg(v) = K for all
vertices v in G. For example, c_9 is a 2-regular graph, because
every vertex has degree 2. For some K greater than or equal to 2,
neatly draw a simple K-regular graph that has a bridge. If it is
impossible, prove why.

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

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...

show that any simple, connected graph with 31 edges and 12 vertices
is not planar.

Show that if G is connected with n ≥ 2 vertices and n − 1 edges
that G contains a vertex of degree 1.
Hint: use the fact that deg(v1) + ... + deg(vn) = 2e

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