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 v, deg v +deg u >=n then G is hamiltonian.

Can you explain it step by step and draw a graph.

Solution Preview

For any n>=3, we construct a nonhamiltonian graph G of order n such that degu+degv>=n-1 for all distinct nonadjacent u and v. The graph is as follows.
First, we select n-2 nodes and construct a complete graph H with n-2 nodes.
Then, we select a node u in H, and other two single nodes v,w not in H.
Then, we set an edge ...

Solution Summary

Hamiltonian and Nonhamiltonian Graphs are investigated.

Explain this problem with a graph to understandand 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

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

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see attached
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Please see the attachment for the full problem.
Consider the particle in a box problem for a box of length ...
Verify the substitution that the solutions ...

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