Is the sequence (4, 4, 4, 2, 2) graphical? Conclusion: it seems it
shouldn’t be, because there are too many vertices of degree 4 and not enough
other vertices of high enough degree to absorb the edges these necessitate.
One theme of this course is that we will be more interested in how we can
adapt the methods of proof to prove related results, and when strengthenings
of results are possible.
If we look at the proof of Havel-Hakimi, all that we needed for the proof to
work was that the vertex vj had a neighbour that vi didn’t have. Initially, we
made an assumption that d(vj ) > d(vi), which guaranteed this for us. However,
we soften remove this assumption, for if d(vj ) = d(vi) and v1 ∈ Γ(vi)\Γ(vj ),
then this also guarantees the existence of such a u.
Exercise 1.25. Adapt the statement and proof of Havel-Hakimi in light of this
observation.

To determine if the sequence \((4, 4, 4, 2, 2)\) is graphical, we need to check if it is possible to construct a simple graph (no loops or multiple edges) that has these vertex degrees. The Havel-Hakimi algorithm is a commonly used method for this purpose.

### Applying the Havel-Hakimi Algorithm
The Havel-Hakimi algorithm works by repeatedly reducing the problem to a smaller one:

1. **Sort** the degree sequence in non-increasing order.
2. **Remove** the first degree \(d_1\).
3. **Subtract 1** from the next \(d_1\) degrees.
4. **Repeat** the process with the new sequence.

Let's apply this step-by-step:

1. **Initial sequence:** \((4, 4, 4, 2, 2)\)

2. **Sort:** \((4, 4, 4, 2, 2)\) (already sorted)
   
3. **Remove the first 4:** We remove the first 4 and reduce the next four degrees by 1:
   \[
   (4, 4, 2, 1, 1)
   \]

4. **Sort:** \((4, 2, 1, 1, 1)\)

5. **Remove the first 4:** We remove the first 4 and reduce the next four degrees by 1:
   \[
   (2, 0, 0, 0)
   \]

6. **Sort:** \((2, 0, 0, 0)\)

7. **Remove the first 2:** We remove the first 2 and reduce the next two degrees by 1:
   \[
   (0, -1, -1)
   \]

At this point, we see negative degrees, which indicates that the sequence \((4, 4, 4, 2, 2)\) is not graphical. There are not enough vertices with sufficient degrees to accommodate the edges required by the vertices with degree 4.

### Conclusion
The sequence \((4, 4, 4, 2, 2)\) is not graphical because the Havel-Hakimi algorithm results in negative degrees, confirming that it's impossible to construct a simple graph with this degree sequence.

### Adapting Havel-Hakimi Proof

Given the observation that we can soften the assumption that \(d(v_j) > d(v_i)\) to \(d(v_j) \geq d(v_i)\) and ensure that \(v_1 \in \Gamma(v_i) \setminus \Gamma(v_j)\), we adapt the statement and proof as follows:

1. **Statement Adaptation:**
   If the degree sequence \(d = (d_1, d_2, \ldots, d_n)\) is graphical, then the sequence obtained by removing \(d_1\) and reducing the degrees of the next \(d_1\) highest degrees by 1 (if they are positive) is also graphical.

2. **Proof Adaptation:**
   - Start with a sorted sequence \(d\).
   - Assume \(d_1 \geq d_2 \geq \ldots \geq d_n\).
   - Remove \(d_1\) and consider the next \(d_1\) highest degrees.
   - Reduce each of these \(d_1\) degrees by 1.
   - Ensure that if two vertices \(v_i\) and \(v_j\) have equal degrees, there is at least one neighbor in the set of vertices whose degrees are being reduced, which \(v_i\) has but \(v_j\) does not.
   - Repeat the process with the new sequence.

By doing so, we are essentially ensuring that there is a vertex whose neighborhood intersects with the set of vertices whose degrees are being reduced, maintaining the graphical property of the sequence.

### Further Study and Practice
To deepen your understanding, here are some related exercises:

1. Determine if the sequence \((3, 3, 2, 2, 2)\) is graphical.
2. Adapt the Havel-Hakimi algorithm for directed graphs.
3. Prove that if a degree sequence is graphical, the sum of the degrees must be even.
4. Determine if the sequence \((5, 3, 3, 2, 2, 1)\) is graphical.
5. Explore the conditions under which a sequence with a zero degree can be graphical.

### Tip
When applying the Havel-Hakimi algorithm, always ensure the sequence is sorted in non-increasing order before removing the largest degree and reducing the subsequent degrees. This helps in systematically checking the graphical property of the sequence.

Is the sequence (4, 4, 4, 2, 2) graphical? Conclusion: it seems it shouldn’t be, because there are too many vertices of degree 4 and not enough other vertices of high enough degree to absorb the edges these necessitate. One theme of this course is that we will be more interested in how we can adapt the methods of proof to prove related results, and when strengthenings of results are possible. If we look at the proof of Havel-Hakimi, all that we needed for the proof to work was that the vertex vj had a neighbour that vi didn’t have. Initially, we made an assumption that d(vj ) > d(vi), which guaranteed this for us. However, we soften remove this assumption, for if d(vj ) = d(vi) and v1 ∈ Γ(vi)\Γ(vj), then this also guarantees the existence of such a u. Exercise 1.25. Adapt the statement and proof of Havel-Hakimi in light of this observation.

Determine if the sequence (4, 4, 4, 2, 2) is graphical using the Havel-Hakimi algorithm. Understand the step-by-step process and proof adaptation to verify the graphical property of degree sequences in graph theory. Suitable for undergraduate students.

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