Prove G contains a cycle of length at least k+1

[cxc001]

This is a graph theory related question.

Let G be a simple graph with min. degree k, where k>=2. Prove that G contains a cycle of length at least k+1.

Am I suppose to use induction to prove G has a path length at least k first, then try to prove that G has a cycle of length at least k+1? Or should I go directly use induction to prove G contains a cycle of length at least k+1?

 

[fzero]

Consider one of the vertices, \nu_k of degree k. Let the set of vertices which are connected to this one be \{ \nu_i^{(k)} \}. What is the minimum path length of a cycle C^{(k)} that connects all of the \nu_i^{(k)}? Can you see a way to construct a cycle C' of degree \text{deg}\, C^{(k)}+1?

 

[cxc001]

Tell me if I'm not on the right track.

Use induction on k.

Pick a path P of maximum length, and suppose vertex vi is a vertex on this path, which has degree at least k, with a set of adjacent vertices {w1,w2,…,wj}, the adjacent vertex set must be on the path.
The minimum path length of a cycle Ci that connected all of vertex set {w1,w2,…,wj } is k.
Then extend the path to a cycle by adding the edge wjvi, so the resulting cycle has length at least k+1.
We're done! Does this induction make sense?

 

[Office_Shredder]

with a set of adjacent vertices {w1,w2,…,wj}, the adjacent vertex set must be on the path.

This is not likely to be true

 

[fzero]

Your path P is too arbitrary and it's unlikely that you can prove very much given the constraints. However, it should be possible to argue that a simple graph contains a cycle that actually contains the vertex set w_i and consider the path length of a closely related cycle that includes v.