The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can be the degree sequence of any graph?
(A) {7, 6, 5, 5, 4, 4, 3, 2}
(B) {7, 7, 6, 6, 5, 4, 2, 2}
(C) Both (A) and (B)
(D) None


Answer: (A)

Explanation: Solve these by Havel- Hakimi procedure:
(I)

7  6  5  5  4  4  3  2 

Remove 7 and decrement by 1 each up to 7 places,

5  4  4  3  3  2  1 

Remove 5 and decrement by 1 each up to 5 places,

3  3  2  2  1  1 

Remove 3 and decrement by 1 each up to 3 places,

2  1  1  1  1 

Remove 2 and decrement by 1 each up to 2 places,

0  0   1  1 

Sort,

1  1  0  0 

So, there are even number of 1\’s, this is valid degree sequence of a graph.

(II)

7  7  6  6  5  4  3  2 

Remove 7 and decrement by 1 each up to 7 places,

6  5  5  4  3  2  1 

Remove 6 and decrement by 1 each up to 6 places,
4 4 3 2 1 0
Remove 4 and decrement by 1 each up to 7 places,

3  2  1  0  0 

No three degree left to decrease, So the graph is not possible for this degree sequence.

Option (A) is correct.

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