Question

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.

Answer #1

Graph Theory
Prove that if G is a graph with x(G-v-w)=x(G)-2 for every pair
of vertices v and w in G, then G is complete.
Hint: assume G is not complete.

10.-Construct a connected bipartite graph that is not a tree
with vertices Q,R,S,T,U,V,W.
What is the edge set?
Construct a bipartite graph with vertices Q,R,S,T,U,V,W such
that the degree of S is 4.
What is the edge set?
12.-Construct a simple graph with vertices F,G,H,I,J that has an
Euler trail, the degree of F is 1 and the degree of G is 3.
What is the edge set?
13.-Construct a simple graph with vertices L,M,N,O,P,Q that has
an Euler circuit...

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.

(a) Let L be a minimum edge-cut in a connected graph G with at
least two vertices. Prove that G − L has exactly two
components.
(b) Let G an eulerian graph. Prove that λ(G) is even.

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 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.

Graph Theory
Let v be a vertex of a non trivial graph G. prove that if G is
connected, then v has a neighbor in every component of G-v.

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?

A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V, E0 ) with the same vertices as G. Show that
every connected graph G = (V, E) contains a spanning tree. (It is
the connected subgraph (V, E0 ) with the smallest number of
edges.)

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.

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