Question

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.

Answer #1

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.

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.

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 u and v be distinct vertices in a graph G. Prove that there
is a walk from ? to ? if and only if there is a path from ? to
?.

Intro to graph theory question:
1) Draw a graph G with w(G) = 2 (w(g) is clique number)
and x(G) = 5 (x(g) is chromatic number)

Discrete math, graph theory question:
Let G be a graph with 100 vertices, and chromatic number 99.
Prove a lower bound for the clique number of G. (Hint: Any lower
bound will do, but try to make it as large as you can.)

Graph Theory, discrete math question:
Let G be a graph with 100 vertices, and chromatic number 99.
Prove a lower bound for the clique number of G. Any lower bound
will do, but try to make it as large as you can.
Please follow this hint my professor gave and show your work,
Thank you!!
Hint: can you prove that the clique number is at least 1? Now
how about 2? Can you prove that the clique number must be...

Prove that every graph has two vertices with the same degree.
(hint: what are the possible degrees?)

Prove that the order of complete graph on n ≥ 2 vertices is
(n−1)n 2 by...
a) Using theorem Ʃv∈V = d(v) = 2|E|.
b) Using induction on the number of vertices, n for n ≥
2.

Graph Theory
.
While it has been proved that any tree with n vertices must have
n − 1 edges. Here, you will prove the converse of this statement.
Prove that if G = (V, E) is a connected graph such that |E| = |V |
− 1, then G is a tree.

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