00:01
Okay, so we have our two degree sequences here.
00:03
We'd like to determine if we can write out a graph that has these degree sequences.
00:09
So for this first part, this part a, i want to remind you of just one of our very familiar graph theory like facts.
00:19
So what we can say is that if we sum over the vertices and add up the degrees of those vertices, this has to be equal to twice the number of edges.
00:33
So namely, what does this tell us? well, this tells us that, so that's the sum of degrees has to be even.
00:49
Well, so let's look at this, what do we have here with this vertex set, this degree set.
00:56
So we have five plus four, plus three, plus two, plus two, plus.
01:01
Plus 1 plus 0, well, this is equal to 5 plus 4 is 9, plus 3 is 12, plus 2 is 14, plus 1 is 15, which is odd.
01:14
So since we said that the sum of the degrees for any graph has to be even, but this degree set gives us an odd sum.
01:25
Well then we can say that this degree set is impossible.
01:38
Okay, so let's look at the second one, though.
01:41
So we're going to follow a construction that you might be familiar with.
01:44
It's called the havel -hakimi theorem.
01:47
So havel, hakimi, just to write that so you can maybe refer, or, you know, like look back at that.
01:56
But, okay, so how does this construction work, though? so we're going to start with, let's start with six vertices.
02:02
One, two, three, four, five, six.
02:06
And then, well, okay, so we know that we want one degree five vertex.
02:10
So let's just connect that one to all the other five vertices.
02:16
And then how we want to think about this is, well, this top vertex, this is all done.
02:20
We don't need to do anything else with that again.
02:23
So now we almost just want to think of this as, well, now we have these five, vertices and we want to draw a graph on them to finish off our original degree set.
02:37
But what does that mean? what would the degree set for those vertices be to match up with that? well, the new, we sort of want to think of this as that now if we can fill the rest in with this degree set, so what i've done is that i sort of got rid of this first one and i subtracted one from each of these, because, which makes sense because we've, each of these vertices has degree one now.
03:03
Well, if we can put a degree set, a graph with the reset this on the rest of this, then it will be good.
03:13
So let's look at that.
03:14
So now, well, let's start with this leftmost vertex and let's connect it...