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05/21/2013 01:35:06 PM · #21226 |
You're very persistent Spork.
Just so you know, I'll keep posting to this thread every day for the rest of my life, or until DPC doesn't exist anymore, or until the internet runs out of space. |
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05/21/2013 01:37:33 PM · #21227 |
Well, I took a couple wild guesses on that round and was way off. Then I asked this guy where I was...
and he did not respond so I knocked him down and stole his bike (and his bag of onions) and rode around the countryside and forgot why I was there.
I am currently sitting by a river chomping down an onion somewhere in Slovakia (I think), winning. |
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05/21/2013 01:39:14 PM · #21228 |
Originally posted by JH:
You're very persistent Spork.
Just so you know, I'll keep posting to this thread every day for the rest of my life, or until DPC doesn't exist anymore, or until the internet runs out of space. |
Spork is eternal.
Spork wins. |
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05/21/2013 01:45:43 PM · #21229 |
spork only thinks he wins
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05/21/2013 02:08:19 PM · #21230 |
| All of you exist because Spork wills it. |
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05/21/2013 02:39:03 PM · #21231 |
| I saw a Spork in the road this morning. ...and ran over it. |
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05/21/2013 03:44:25 PM · #21232 |
Spork is not that spork.
Spork wins |
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05/21/2013 04:08:54 PM · #21233 |
Awww what the heck did I lose anyways
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05/21/2013 04:21:44 PM · #21234 |
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05/21/2013 04:39:04 PM · #21235 |
Originally posted by tolovemoon:
Awww what the heck did I lose anyways |
and we lost the last 4 years of your life. welcome back. ...but I win. :) |
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05/22/2013 12:52:12 AM · #21236 |
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05/22/2013 08:23:24 AM · #21237 |
| What, you lost 8 months of your life? |
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05/22/2013 08:34:04 AM · #21238 |
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05/22/2013 08:34:54 AM · #21239 |
Sporl knows where it is, because Spork hid it.
Spork wins. |
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05/22/2013 08:45:40 AM · #21240 |
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05/22/2013 08:50:09 AM · #21241 |
Sporl is Spork's minion...Just like Sporn
Spork has minions...Spork wins |
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05/22/2013 01:22:34 PM · #21242 |
Originally posted by bvy:
Originally posted by bvy:
My most recent exploration into graph theory involves vertex colorings. The problem du jour is stated as follows: Let G be a k-chromatic graph and r = D(G). (Here D(G) signifies capital delta of G, the maximum degree of any vertex of G.) Show that there exists a k-chromatic r-regular graph H, which has G as an induced subgraph.
Assuming that G itself is not a regular graph (the trivial case), there is some vertex v which has degree strictly less than D(G). To "regularize" this vertex, v must add exactly D(G) - deg(v) edges to vertices of H which are not also vertices of G. This must be done for all vertices of G having degree less than D(G).
So constructing H is a matter of adding new vertices and edges which regularize all of the "smaller" vertices of G. The two caveats to this are: one, the new vertices of H must each have degree exactly D(G), and two, we must not add edges in a way that increases the chromatic number of H beyond that of G.
Thoughts or insight? |
Thanks for your help, everyone. As promised, here's a picture. As an example, I'm using the graph G=T(4,1) since it has vertices of degree 1, 2 and 3. It has chromatic number two as shown by the red and green vertices. To construct H, I partitioned its would-be vertices by color -- necessarily, red and green. This picture is the first step, and I will complete it later.
The picture shows only one edge going to a green vertex of H. So if H is to have a green vertex, and be 3-regular, it needs two more edges. Those will necessarily be adjacent to red vertices in H. Do you see where this is going? |
Here's the completed diagram showing, for the graph T_4,1, a 3-regular, 2-chromatic proper supergraph. The strategy of partitioning the vertices of H\G by the colors of G was an interesting one and offered some visual insight to the problem for this one graph. But in order to show that it's true in general, a quite different strategy is required. I'm still working out the semantics of the proof.
Thanks again for your support, everyone. I'll follow up soon. |
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05/22/2013 01:29:34 PM · #21243 |
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05/22/2013 02:44:12 PM · #21244 |
Originally posted by JH:
What, you lost 8 months of your life? |
Not really 8 months. But I've participated in the thread for 8 months (loser me), not every day all day long though. |
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05/22/2013 03:07:33 PM · #21245 |
Originally posted by beatabg:
Originally posted by JH:
What, you lost 8 months of your life? |
Not really 8 months. But I've participated in the thread for 8 months (loser me), not every day all day long though. |
You say that like there's something wrong with it. :( |
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05/22/2013 03:13:19 PM · #21246 |
Originally posted by Art Roflmao:
You say that like there's something wrong with it. :( |
ABSOLUTly not. Is that what it sounded like? Not intended at all :P Would you like to talk about it? Just proceed ;) Haha |
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05/22/2013 03:23:00 PM · #21247 |
Ok, once again I will solve  bvy's math problem...
The box needs to be flatter, more pepperoni on the top pizza, NO PICKLES on any pizza and Delivery in thirty minutes or less.
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05/22/2013 03:29:50 PM · #21248 |
Spork now orders 37,593,923,478 pepperoni pizzas to feed himself and his legion.
And they'd better be here in 30 minutes or less, or Spork will be angered.
Spork wins. |
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05/22/2013 03:30:18 PM · #21249 |
| Would you like fries with that? |
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05/22/2013 03:31:10 PM · #21250 |
Originally posted by Art Roflmao:
Ok, once again I will solve bvy's math problem...
The box needs to be flatter, more pepperoni on the top pizza, NO PICKLES on any pizza and Delivery in thirty minutes or less.
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I didn't even feel like looking at the problem. I've been studying so much lately that I feel like my brain is gonna explode. I can find a medical reference in almost every sentence now and the remaining of my sanity is in jeopardy. Taking a break Saturday - then I'll read the pizza math problem. |
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