Question

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.

Answer #1

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

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.

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?

please solve it step by step. thanks
Prove that every connected graph with n vertices has at least
n-1 edges. (HINT: use induction on the number of vertices
n)

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...

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.

Prove that a bipartite simple graph with n vertices must have at
most n2/4 edges. (Here’s a hint. A bipartite graph would have to be
contained in Kx,n−x, for some x.)

Exercise 10.5.4: Edge connectivity between two vertices.
Two vertices v and w in a graph G are said to be
2-edge-connected if the removal of any edge in the graph leaves v
and w in the same connected component.
(a) Prove that G is 2-edge-connected if every pair of vertices
in G are 2-edge-connected.

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.

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.

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