Question

Show that a graph is connected if and only if there is no bipartition of the set of its vertices such that no edge has an endvertex in each subset of this bipartition.

Graph Theory

Answer #1

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...

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.

Show that an edge e of a connected graph G belongs to any
spanning tree of G if and only if e is a bridge of G. Show that e
does not belong to any spanning tree if and only if e is a loop of
G.

Show that an undirected graph G = (N,A) is connected if and only
if for every partition of N into subsets N1 and N2, some arc has
one endpoint in N1 and the other endpoint in N2.

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

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.

(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.

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.

Prove that your conjecture will hold for any connected
graph.
If a graph is connected graph, then the number of
vertices plus the number of regions minus two equals the number of
edges

Suppose that H is a connected graph that
contains a proper cycle. Argue that removing any single
edge from this cycle will leave a subgraph of H
that remains connected.
Make sure you are fully addressing the technical
definitions involved --- do not just talk vaguely about
vertices being connected, you need to discuss specific paths
between vertices.

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