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The graph G is planar.The graph G has chromatic number four_The graph G has radius threeThe graph G has diameter three...

Question

The graph G is planar.The graph G has chromatic number four_The graph G has radius threeThe graph G has diameter three

The graph G is planar. The graph G has chromatic number four_ The graph G has radius three The graph G has diameter three



Answers

The thickness of a simple graph $G$ is the smallest number of planar subgraphs of $G$ that have $G$ as their union.
$$
\begin{array}{l}{\text { Show that if } G \text { is a connected simple graph with } v \text { ver- }} \\ {\text { tices and } e \text { edges, where } v \geq 3, \text { and no circuits of length }} \\ {\text { three, then the thickness of } G \text { is at least }[e /(2 v-4)]}\end{array}
$$

Where has to show that G is a connected, simple graph with reverted season E edges where the number of Vergis eases at least three. Then the thickness of G is going to at least be these ceiling of E over three V minus six. So let G B A connected simple graph with the Vergis ease and E edges where V is going to be greater than or equal to three now by corollary. One from the book follows that the number of Edges E is less than or equal to three V minus six. This is why we require that he's greater than they were three. And since the graph has v verse, ease and edges, this means that we need at least e over three V minus six plainer graphs. He drove all edges without crossing any edges and then by the definition of thickness. This means that the thickness of the graph of G needs to be at least e over three V minus six. And of course, since the thickness is an integer, it follows that thickness of G must be greater than or equal to the ceiling of eat over three V minus six

Hey, guys. And this problem, we're going to be grant from the equation G of X equals three of 3/4 to the X power. Um, this function were denoting is G of X. In another video, I've stated that G of X could be any we can call this dysfunction here. Anything in this case, we're gonna call it G of X, Whereas we could call it F of X. Why? But it will not affect the behavior of this function. So let's get started. We have in the X y table, we're gonna test the value Negative one first. So 3/4 to the negative first power. That's just gonna be the reciprocal, which is 4/3. When X is equal to zero, we have G of X is equal to 3/4 to 0 power. Anything to the zero power is going to be one, three or four to the first power. That's just going to be itself. Anything to the first power is itself. When X equals 2 3/4 to the second power that will be simply three squared, divided by four squares. So that will be 9/16 And similarly, when X is equal to three, we have three Q divided by four cubes. So three times nine that's 27 divided by four times 16. That is 16 4. So let's go ahead and graph it. Why X negative x get a little nicer. So at X equals negative one Y is equal to 4/3. So right about here, when X equals zero, why is one right here when X equals one wise 3/4 when X equals two, why is 9/16? So a little bit bigger than half. So this would be half Yeah, a little bit bigger. And and I forgot to write this over here, that would be 2764. So when X is equal to three, this is 27/60 for a little bit less than half. So this is still an exponential function, but it definitely is increasing a lot slower and approaching the X axis a lot. So little racial

In this question were given the function G of X equals 1/2 times X minus 3/2 squared plus 1/4 and or given the fact that X equals one excuse me, X equals 1/4 is the axis of symmetry and would want to determine. Is that true or not? Well, the axis of symmetry goes through X equals h the value that we're subtracting from X and so applying that to this function the accident symmetry should be located at three halves. And so we can also I confirmed this graphically for look like decimals. We've got one hat in the given function. It's in red, and also X equals 1/4 asking. Is this the axis of symmetry? Well, this problem is not symmetric about this line. It should go through, uh, 1.5 or three halves, as we stated here. So the answer to this question is a big false

We have to Graft G of X equals negative three. So we had this here with negative one negative to negative. Three R g of X is just gonna be a flat horizontal line right here. Negative three. Where? At every point along this The Y value is night of three.


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