5.4.2.1. Case 1: Vertex v is Adjacent

Today, we're starting with degree sequences. A degree sequence is simply the list of degrees of the vertices in non-increasing order. Can anyone tell me what this means?

It means we start with the highest degree and list them down until the lowest?

Exactly! And we denote this as a sequence S. A key point to remember is that for a sequence to be a graphic sequence, it must represent a simple graph. What condition can you think of that must be satisfied?

The sum of the degrees must be even, right?

Right! That is essential for any graph, not just simple graphs. Great point. The reason is that each edge connects two vertices, contributing two to the total degree.

What if there are negative degrees?

Good question! Degree values must always be non-negative. Negative degrees aren't possible because you can't have a vertex 'losing' edges. Remember, a vertex with a degree of zero has no connections.

So, if we had a sequence like 5, 4, 3, 2, 1, 0, it wouldn't work for a simple graph because one vertex is zero?

Correct! If we have six vertices, vertex with degree 5 must be connected to five others, leaving none for the vertex with degree zero. This makes it impossible for such a degree sequence to represent a simple graph.

Let’s summarize. A degree sequence lists vertex degrees in non-increasing order, must sum to even, and cannot include negative values.

Let's apply what we've learned with the Havel-Hakimi theorem. Can anyone explain what this theorem involves?

It says that you can create a new sequence by removing the top degree and reducing the next few degrees.

Exactly! We take the highest degree, let's call it d, and remove it from the sequence. Then we subtract 1 from the next d degrees.

What does the new sequence look like?

The reduced sequence, S*, still needs to be in non-increasing order. Can anyone give an example of reduction?

If we had a sequence 5, 3, 2, 2, 1, we’d remove 5 and reduce the next three by 1. That would give us 3, 1, 1, 0?

Yes, that’s a great application! If S* is also a graphic sequence, then S is too. However, if S* fails to be graphic, neither does S.

What happens if we keep reducing until we reach a trivial case?

If you end up with a small sequence that is easily verifiable, you've simplified the process significantly. Always remember to check until you reach a manageable size!

So to summarize this session: The Havel-Hakimi theorem allows us to systematically determine if a degree sequence is graphic by reducing it.

Now, let’s verify some sequences. First, is the sequence 6, 5, 4 valid?

The sum is odd, so it can't be graphic.

Correct! Here's another sequence: 3, 3, 3, 3, 3, 3. What do you think?

All degrees are equal, and it sums to 18. I think it can represent a hexagon!

Absolutely! It's a perfect example of a graphic sequence.

What about 5, 4, 3, 1, 0?

Good observation! Let's see. Vertex with degree 5 cannot have a degree 0 neighbor. Therefore, it's not graphic.

As we wrap up, verify sequences by checking their total sum and adjacency possibilities.