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This is a practice question from the text.

The Question : Show that a tree with two vertices of degree $3$ must have at least four vertices of degree $1$. I have the answer to PART A.

Part B) Show that the result of Part (a) is the best possible i.e. a tree with two vertices of degree $3$ need not have five vertices of degree $1$.

I would really appreciate a hint that could help me solve part B.

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For Part B, consider this tree: >-<

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  • $\begingroup$ thanks for the reply. So should my proof be that the tree >-< is the simplest form of a tree with two vertices of degree 3 and it clearly has 4 vertices of deg 1. $\endgroup$ – Kj Tada Jun 5 '13 at 23:33
  • $\begingroup$ You only need one example to show (a) is best possible, and this does the trick. Others will too, e.g. >----< $\endgroup$ – vadim123 Jun 5 '13 at 23:34
  • $\begingroup$ thankyou again, that is very intuitive however is there a way to prove that the result in Part a is the best possible. As just sketching the graph >-< only shows that result of Part a is possible but not the best possible. I could be over thinking this. $\endgroup$ – Kj Tada Jun 5 '13 at 23:39
  • $\begingroup$ I think your difficulty stems from understanding what it means that "the result in Part a is the best possible". You have proved a lower bound on something (namely, the number of vertices of degree 1 in a tree with two vertices of degree 3 is at least 4). If this were not the best possible, that means that there would be a better lower bound, i.e. a higher lower bound. A higher lower bound would be at least 5. But 5 can not be a lower bound: >-< is a counterexample. $\endgroup$ – Mees de Vries Jun 6 '13 at 0:02
  • $\begingroup$ Thank you you were right I was having a hard time understanding the fact that it is best possible and your explanation helped. Thanks vadmin123 and Mees de Vries $\endgroup$ – Kj Tada Jun 6 '13 at 0:24

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