Okay, so what's the graph from what's a plane?

And what's embedding?

I'll tell you all about that photograph consists of a bunch of foreign skeptical verses, and then some pairs of Earth's is air joined by a line that tickle and edge.

And like this and Sampras averts is don't have anything between them.

So, for example, this is a graph.

This is also Graf called the house you actually call that helps we actually go over the house way.

Also, its official name is B five.

Confident its nickname is helps.

Professor.

When I was school, a graft may had, like an X and A Y access and the line and things like that.

But this is not what that looked like.

Completely unrelated is just an unfortunate, uh, overloading, overworked lately, different concept.

It'll Okay, So this is what I think.

This is a real graph.

Yes, exactly.

There we go.

Thank you.

Somebody for somebody who understand.

Okay.

And then the question is were asking your drawer.

So this is a plain piece of paper is a plane.

And the question is which grafts can you draw in the plane s O that ages don't intersect.

So what does me natures don't understand Don't intersect where they should, for example, in these graphs, the sage and they said the Intersect over here, but you meant it.

There intersect over here because they have a common rex over here.

But for example, if I make these drawing so it's all the verses are 1234 The age of the 123 These age and their page the Intersect.

And it's not a fair intersection.

They just intersect because I drew it like this.

I could have drawn it like that.

So graphics plainer if you can join in the plane or in a piece of paper in such a way that to wages on the Intersect, where you intended them to intersect only in the second it works.

Professor, if you draw a graph and edges intersect, does that mean it's not a graph or it's invalid?

Or it's like breaking the rules?

Or is it just a different kind of graph?

It's a different kind of graph.

It's so a graph.

This is a graph in this photograph it's called an abstract graph, and then you can ask about We'll talk about a plane and embedding autograph and what that man said that you've thrown it like this.

This is a graph but not playing that.

I'm betting this is only illegal if what you're after, it's plain.

Embedding is a plainer embedding.

Somehow better it is.

It's probably possibly good for some applications, like if you want to in a V L a psych component or something which I don't know anything about.

But it's just it's just another appropriate and math.

We'll do it.

We'll think of an object to think of a property and then we say, Do all the soldiers have this property?

Though some of the Soviets has this property, can you tell if you have this property?

What are the property?

Do you have?

Can you characterize when you have this property so being plainer, it's a property some graphs have and sound graphs don't have.

And I actually haven't told you what plane it means yet.

I think I told you what a plane they're embedding is, But I haven't told you what the plane a graph is so graphics plainer if it has a plane there embedded, for example.

These Graff has a plane there.

Embedding namely that one.

These guys also has a plan embedding name to this one.

This graph, no matter how hard you tried, does not have a plan.

I'm betting you couldn't cheat.

I will manipulate.

Move.

That is exactly that is a direct witness.

Here's an example.

It looks almost almost as complicated.

That right?

These only different from bed by one drinks.

But these you could it actually in bed in the plane.

Because what you do issue number, the virtues that you believe me, it's a embedding.

And then you do this.

So let me explain.

Why at least is the same with it.

What do you care about?

Care about Versace.

And we care about which pairs of verses are adjacent.

So verses I want 2341234 That's fair.

Now here, every pair is adjacent.

So let's change it.

Also hear every pair is adjacent.

Wanted Jason to do 22332 1/4 address into the mall.

So these abstract graph is the same as that abstract stuff.

This is plain embedding this is not a plane.

Edges have to be straight like, Could you have kept these voter sees in the same places and, like, drawn curved lines around the place toe as well?

Or I just don't have to be stating the terms that said that anything you can embed with curved ages, you can embed with straight ages.

It doesn't matter.

Uh, so you're asking me, Could I leave these four verses in place and just, uh, make that just yet?

So what I would do is so I could make this completely s symmetric ugly drawing or this beautiful join?

Yes, less elegant.

But on the other hand, you know, they say, you can see how to get it, which I think has a lot of value.

So it's not a conjecture.

It's a theorem that any plane a graph that you can draw with curved edges.

You could also have demonstrated that That's really cool.

Yeah, it is amazing.

I learned a Stearman the right time in my life.

I was in high school when I heard about it.

So I am, in fact, as amazed by the stream as one should be.

I haven't become jaded by the time I heard of it.

But you can't do any trickery with that like you did there.

That one's trick there is one more week and doing a trickery, and in fact, that's usually how interaction to plan a graph start.

So you have three houses and three utilities.

Those houses of actual houses.

They're not these houses.

I don't know.

You tell me.

No way to tell.

You want to put lines between the utilities and the House is right.

You want to put power lines and phone lines and water pipes, and it's important that they don't intersect each other so that you can fix fix them independently or some other story along this line.

And so the question is, Can you That's all you know, you can try.

You can never connect this one like that.

So you got the electricity and you can connect the form and we can go here.

Maybe you can go there.

May be could go there, and you can start with the water, but you're not gonna be able to finish, and, uh, and the city and they're not going be able to figure.

So if I drove, this is a graph.

Everybody on the first side is adjacent to everybody on the second side.

And this is called case history.

By the way, these guys called K fights.

Okay, five in cases, three are two graphs that do not have planning buildings there.

The two smallest graphs don't have plan on being okay.

So what?

Uh, mystifyingly stands for complete.

And so this is a complete graph on five verses.

Now, this is not the complete graph, because obviously there's some pairs of their non adjacent.

You have two sides, this side and that side, and everybody on this side is adjacent.

Everybody on that time, that's called the Complete by protective.

And the way we know that it's complete Viper tight and not complete is because we have two numbers instead of one number.

So Kay bee's a here be there all adjacent to each other.

Katie's de OlPerez address.

Is there a way to tell what's invaluable and what's not?

So there's a way to tell, and actually, that's a lovely tear with him.

I can tell you you're really committed to Canton bed.

That's in fact, maybe you shouldn't believe me, but there is an elegant proof of it that you need to know a little bit of German.

You need to know something called Oilers formula.

And from that that's actually very easy to see that these two graphs or not incredible.

But let's say we believe that they're not metal.

So then here's another graft.

It's not incredible if I take an edge here and I won't recall subdivided, which means I'll draw some verses in the middle.

That's another graft.

It's not incredible, right?

Because if I couldn't bet this now, just delete the verses.

And so when you can do that for any a jeweller here, there.

So this is all not unbeatable, and they're called subdivisions.

So we know that Kay five is not in available cases.

Three is not in available, and we also convinced ourselves that subdivisions off them little Milton but and the team is, that's it.

The only reason some graph is not incredible is because there's one of those bad guys sitting in.

You can delete Roaches in ages from it and find either subdivision of case.

There's your subdivision of K five.

It's called Berezovsky's.

Just those two.

Just those two.

It's a medical so They're like the two spanners in the works.

Exactly.

So the term is G is cleaner.

If if and only if G has no Okay, five subdivision and no.

Okay.

Wow.

So every single grab If I drew you the most complicated graph in the world and it was known in vegetable, you'd be able to find that in there somewhere.

That is correct.

That's the only reason something is not about so.

This is a very special what more scraps are.

Not likely.

It's a very special kind of graphs that are incredible.

It so that other graphs and other graphs Abed autograph so good.

In fact, most of the workers without a craps is justice.

If you work with those, you can suddenly prove much stronger terms.

It's like, um, um, you know, you can think of all numbers or you can think of powers of two are a problem.

Prime numbers, powers of store Much nicer than the old numbers.

It's know that the other numbers are not good.

This is just, you know, in some way that the numbers and more interesting because they're no parts of two.

But they're certainty.

Ramsey can only provable powers of two.

So probably the most famous terrible hearing about Elena graphs is that you can color them with four colors.

Some rebel talk about that little bit so calorie and Brafman's You want thio color the versus so that a Jason forces get different colors.

So, for example, this graphic in color this blue.

This is green and also this green this red and just like this blue, and this is a coloring.

This is calling with three colors.

Jason bask in different colors, this coloring with three colors, and you can see that you couldn't do it with fewer colors in three.

Because you have these super total Perez adjacent, so you have to use it.

And now the question is, if I give you a graph, what's the smallest number of colors you can use to color it?

So one thing you mean it largest is that if you have a bunch of words, is all fair right adjacent than, uh, you need atleast that many carbs and said some things.

We need many more.

This is a whole different branch of mass.

It's very active at the moment, but often your union many more.

But at least this isn't always lower bound.

Now, in later graphs, we know that you don't have a K five, so you're not gonna have five verses au pair wise adjacent.

So at least this naive bound only would give you that you need for when the time is four is in their minds.

It's called before quality.

Um, every plane a graph is for colorable.

It comes from Met coloring.

It was actually conjectured Bye bye, cartographer.

And in 18 something it in 63 Think hey said if I have a map and I want to color the country's so that Jason countries get different colors, how many colors do I need?

A.

Seemed to be noticing that four is always enough.

And ah, and the answer is yes.

Four is always enough because you can easily translate coloring a map into Colin Flynn.

A graph.

What you do is take a map, right?

So the country's air this outside dream and then those strangles.

So now I'm going to make a graph out of it.

I'm going to put a vortex in the middle of which country if to come to share the border, I'm going to put an edge between the tourists.

Another way to think about it.

I'm considering the age.

Is this a juice to go like this between these two countries?

But now I'm gonna put it pretty clearly.

So because this is a mess, let's just draw the blue Graf again.

This is already better, but now conductive and better.

You can do that.

So anyway, so the black thing there was a map.

The blue thing.