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In class. We used counting argument to prove that K33 is nonplanar. Modify the argument appropriately to prove that the Petersen graph (Figure 2.6, p.39) nonplan...

Question

In class. We used counting argument to prove that K33 is nonplanar. Modify the argument appropriately to prove that the Petersen graph (Figure 2.6, p.39) nonplanar. Give an example or explain why none exists_ If giving an example, be sure to justify why vour example satisfies the given criteria: planar graph with degree sequence 403),5(),6(3) , 7(3) planar 4-regular graph nonplanar A-regular graph K , Suppose T is a (ree)offorder (a) Find the order and size of the graph T + T. Express YOur an

In class. We used counting argument to prove that K33 is nonplanar. Modify the argument appropriately to prove that the Petersen graph (Figure 2.6, p.39) nonplanar. Give an example or explain why none exists_ If giving an example, be sure to justify why vour example satisfies the given criteria: planar graph with degree sequence 403),5(),6(3) , 7(3) planar 4-regular graph nonplanar A-regular graph K , Suppose T is a (ree)offorder (a) Find the order and size of the graph T + T. Express YOur answers in terms of k. Include couple of sentences explaining the reasoning behind your formulas (b) Prove that if k23 then T + T is nonplanar Prove that there is no connected planar graph that is 4-regular and has seven regions_ (Note: Your proof must work for arbitrary number of vertices S0 in your proof, let n denote the number of vertices:



Answers

Suppose that a connected bipartite planar simple graph has $e$ edges and $v$ vertices. Show that $e \leq 2 v-4$ if $v \geq 3$ .

Okay, so we know in a graph that each Vertex must have a different color right now. Call me a graph with six or fewer verses, while requires six or fewer colors. Let's go ahead. And let's assume that all graphs with K mercies can be colored with six cups. Now we consider a graph G with all K plus one verses. No G has a graph. Um, o G has a vertex honestly register for text V with at most degree six suffering. Remove V to form the graph G prime. Dingy Prime has Cavor sees? Um, and we require, well, six colors. If the neighbors of the do not use oh fire colors, then we use, um um the um 5 to 5 colors. Right? So if the neighbors RV do not use all five colors than we use five colors of G by assigning to beat a color not used by any of its neighbors, so therefore, every player graph G can be colored using six or fewer colors

So with this problem, we're going to have a bi partite graph so it will be partitioned into two sets will have V one and V two. And what this tells us also is that there's an odd number of urgencies. So the all the Vernon season here, plus all the vergis ease and here is an on number. So we know that all of the Verte season V one have to only connect to Vergis is in V two and the other way around. So since there's two sets, they have to contain que elements. Um, and since since each set has K elements, um, that means that it's K plus K omens, which equals two K um, and two K is an even number. So it will only have a Hamilton circuit. If there's an even number of urgencies set. Tells us that with an on number of urgencies, which we're told in the beginning, um is not possible toe have a Hamilton circuit

Right? Yeah. Simple 14 Or were asked to show that the number. What number? My former. No, we have no circuits. Therefore Oh, are you greater than five? By therefore it follows that to remember greater than five women were reaching. It's a way of Berries now. Well, it's for me Are what to so bodies you get my stream Plus two is less. It's a or, in other words, less too. Five, you know, finally is equivalent to he is less than in two live my from

Were given a connected by part tights. When you're simple, graph and his and morass show that number edges lesson. Sometimes Number 30 sees my sport number Gordon seasons at least purpose, but G affected. Like her simple graph thievery sees. Yeah, his yes is greater than reports of three. Now let the very sexy Jean petition he sets they want you to. His G isn't quite partite grand, it seems to my party crack and only have surface of even one and see why plays because of their Texan one. They need to use an even number of edges and, uh, whatever it takes. And do you want again for each time you crossing to cross back into one as the virgin season view on the only connected, diverse Season two and vice visit. So to summarize or using even number, then just that her text you want again. So it follows that my party graph back or graph G does not have a circuit of like three. Therefore, by corollary, three was true. Previous exercise follows that number edges e is lust, sometimes the urgencies being


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