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01/13/2011 02:52:53 PM · #4701 |
| *crumple crumple rip shred chomp chomp* |
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01/13/2011 06:08:11 PM · #4702 |
| so...I guess I'm still winning... |
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01/13/2011 07:25:03 PM · #4703 |
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01/13/2011 07:26:42 PM · #4704 |
| We all know I'll win, nice try Art. |
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01/13/2011 07:27:11 PM · #4705 |
| what are you two talking about!!! I WIN...no need to respond |
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01/13/2011 07:30:20 PM · #4706 |
| If I ever waited for a "need" to respond, I would disappear from this site completely. :P |
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01/13/2011 07:56:22 PM · #4707 |
| You do have a point Ken... |
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01/13/2011 07:59:20 PM · #4708 |
Must....resist....
The needs of the many outweigh the needs of the few. Or the one. |
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01/14/2011 01:56:46 AM · #4709 |
| resistance is futile. bow to your superior "Last wins" competitor. |
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01/14/2011 02:30:01 AM · #4710 |
| this is not some silly square dance...there will be no bowing! |
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01/14/2011 02:43:02 AM · #4711 |
| Damn. Then I better change my shoes. |
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01/14/2011 02:49:54 AM · #4712 |
How about bowling? I'm sure Ken wouldn't mind being a pin.
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01/14/2011 02:57:28 AM · #4713 |
| Alllllrighty. Now I have to go change my shoes once again. |
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01/14/2011 08:22:21 AM · #4714 |
Good morning. A planar graph has a vertex of degree less than or equal to five.
Proof: The result is obvious for planar graphs with six or fewer vertices since the maximum degree of any vertex is five.
Let G(p, q) be a planar graph with p >= 7, and suppose G has no vertices of degree less than or equal to five. Then every vertex of G has degree six or greater. The sum of the degrees of the vertices is 2q which must be greater than or equal to 6p, since each of p vertices has degree six or greater. So 2q >= 6p or q >= 3p. However, a corollary of Euler’s formula (p – q + r = 2 where r is the number of regions of a planar graph) states that q <= 3p – 6. Combining these results, we get 3p <= q <= 3p – 6, or 3p <= 3p – 6, which is a contradiction. Hence, G must have some vertex of degree less than or equal to five.
We will use this result later.
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01/14/2011 09:04:52 AM · #4715 |
| oh, good grief Brian...it's just way to early for this....I will have to say "I win" again |
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01/14/2011 01:00:57 PM · #4716 |
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01/14/2011 01:35:50 PM · #4717 |
| Well I do too...maybe we should all meet there...I win |
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01/14/2011 05:12:46 PM · #4718 |
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01/14/2011 06:35:52 PM · #4719 |
| yes...I do win again...thanks Art... |
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01/14/2011 06:45:21 PM · #4720 |
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01/14/2011 07:02:57 PM · #4721 |
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01/14/2011 07:07:40 PM · #4722 |
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01/14/2011 07:08:23 PM · #4723 |
| stomps foot!!!! I WIN...NO NEED TO RESPOND....HUMF |
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01/14/2011 07:10:46 PM · #4724 |
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01/14/2011 07:35:39 PM · #4725 |
| damn it...I win...now shoooo |
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