(This question has nothing to do with colorings.) Suppose a graph G has a Hamilton path consisting of vertices x1, x2,..., Xn. Conside any induced subgraph G' formed by removing a single vertex from G (i.e., G' is formed from G by removing a single vertex and all edge with that vertex as an endpoint.) Prove that G' has at most two connected components, i.e., the vertices of G' can be partitioned into two sets V and V, such that there is a walk between any two vertices in V and a walk between any two vertices in V2. (Hint: You can define V and V2 using the Hamilton path. In some cases V or V2 can be empty, in which case G' is connected.)

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A:

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A: SUMMARY: - hence, we discussed all the points.

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A: Answer :

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A: ANSWER:-

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A:

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A: Actually, given question regarding graph.

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A: Below i have answered:

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