Question

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 activity you hopefully decided that a graph as described in the activity must:

- have exactly two connected components; and
- each connected component must be a complete graph.

Prove these two statements.

*Notes:*

- Write a separate proof for each of the two statements.
- Your proofs should be general proofs; they should
*not*rely on examples. - Be aware that there isn't just one unique graph G as described in the activity for each value of n. In fact, for each n≥4 there will be n/2 different versions of such maximal simple nonconnected graphs (rounding n/2 down in case n is odd), and the different versions will have different numbers of edges while still having the same number of vertices.

*Hint:* For each of the two proofs, you might consider
arguing by contrapositive or by contradiction.

Answer #1

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?

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.

Let G be an undirected graph with n vertices and m edges. Use a
contradiction argument to prove that if m<n−1, then G is not
connected

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.

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 = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.

Let G be a simple graph with n(G) > 2. Prove that G is
2-connected iff for every set of 3 distinct vertices, a,
b and c, there is an a,c-path
that contains b.

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).

Show that if G is a graph with n ≥ 2 vertices then G has two
vertices with the same degree.

30. a) Show if G is a connected planar simple graph with v
vertices and e edges with v ≥ 3 then e ≤ 3v−6.
b) Further show if G has no circuits of length 3 then e ≤
2v−4.

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