Explain this problem with a graph to understand and explain it step by step.

a) Show that if G is a 2-connected graph containing a vertex that is adjacent to at least three vertices of degree 2, then G is not hamiltonian.

b) The subdivision graph S(G) of a graph G is that graph obtained from G by replacing each edge uv of G by a vertex w and edges uw and vw. Determine, with proof, all graphs G for which S(G) is hamiltonian.

Solution Summary

Hamiltonian Graphs and 2-Connected Graphs are investigated. The solution is detailed and well presented.

Is it TRUE or FALSE that ( and why )
In an undirected graph(with no self loops), if every vertex has degree at least n/2, then the graph is fully connected ?
Thanks

4.15 Show that this theorem 1 is sharp, that is, show that for infinitely many n>=3 there are non-hamiltonian graphs G of order n such that degu+degv>=n-1 for all distinct nonadjacent u and v.
Can you explain this theorem,please
Theorem1: If G is a graph of order n>=3 such that for all distinct nonadjacent vertices u and

Let G be a graph. The line graph L(G) of G is defined to be the graph whose vertices are the edges of G and where two vertices of L(G) are adjacent if the corresponding edges of G are adjacent. Prove that if G is connected, then L(G) is eulerian if vertices of G are all odd or all even.

The solution below got cut off. Please let me know what is the total solution:
problem:
The number of strongly connected components in a graph G is k. By how much can this number change if we add a new edge?
solution:
If we add an edge to a biconnected graph with k strongly connected components, then the

Plot the graphs of the following and identify the graph that represents the cooresponding functions and justify your answer. Thank you!
1) y=2^x
2) y=log 2x (the 2 is subscript)
3) f(x)=6^x
4) f(x)=3^x-2
5) f(x)=(1/2)^x

What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphi

Using the theorems from Graph and Digraphs 4ed by G. Chartrandand L. Lesniak:
1 - (a) Let G a graph of order n such that deg v greater than or equal to (n-1)/2 for every v element of V(G).
Prove that G is connected.
(b) Examine the sharpness of the bound in (a).
2 - Prove the every graph G has a path of length sigma

A graph is outerplanar if it can be embedded in the plane so that every vertex lies on the boundary of the exterior region. Prove the following:
If G = G(p, q) is outerplanar with p >= 2, then q <= 2p - 3.