Question

. Find a bipartite graph with degree sequence (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2), or explain why such a graph cannot exist.

Answer 1

Such a bipartite graph cannot exist.

Step-by-step explanation:

A bipartite graph is characterized by its vertices being divided into two disjoint sets, and any edge connects only vertices from different sets.

given degree sequence (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2) indicates the number of connections each vertex has. In a bipartite graph, the sum of degrees of vertices in each set must be equal since the graph is bipartite, yet this sequence doesn't allow for such equal sums.

With six vertices having a degree of 4, creating pairs in a bipartite graph requires each vertex to connect to four other vertices in the opposite set. However, the remaining vertices (with degrees 6 and 2) do not facilitate an equal distribution in the second set to accommodate these connections.

Moreover, bipartite graphs necessitate an even sum of degrees for both sets of vertices. In this sequence, the sum of degrees results in an odd number (38), making it impossible to divide the vertices into two sets with equal total degrees.

Thus, the lack of balance in degrees within the given sequence prevents the construction of a bipartite graph, confirming the impossibility of such a graph's existence.

Answer 2

A bipartite graph with the given degree sequence does exist.

To determine whether a bipartite graph with a given degree sequence exists, we can use the Havel–Hakimi algorithm. The algorithm proceeds as follows:

1. Start with the given degree sequence.
2. Order the degree sequence in non-increasing order.
3. Remove the first element, say d, and subtract 1 from the next d elements in the sequence.
4. Repeat steps 2 and 3 until the sequence becomes all zeros or it is not possible to continue (i.e., a negative degree is encountered).

If the process terminates with a sequence of all zeros, then a graph with the original degree sequence is feasible; otherwise, it is not.

Let's apply the Havel–Hakimi algorithm to the given degree sequence (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2):

1. Sorting the sequence: (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2)
2. Applying the algorithm:
   • (5, 5, 3, 3, 3, 3, 3, 3, 1, 1, 1) (Remove the first element 6 and subtract 1 from the next 6 elements)
   • (4, 4, 2, 2, 2, 2, 2, 2, 0, 0, 0) (Remove the first element 5 and subtract 1 from the next 5 elements)
   • (3, 3, 1, 1, 1, 1, 1, 1, 0, 0, 0) (Remove the first element 4 and subtract 1 from the next 4 elements)
   • (2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0) (Remove the first element 3 and subtract 1 from the next 3 elements)
   • (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) (Remove the first element 2 and subtract 1 from the next 2 elements)
   • (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (Remove the first element 1 and subtract 1 from the next 1 element)
   • (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (The sequence becomes all zeros)

Since the algorithm terminates with a sequence of all zeros, a bipartite graph with the degree sequence (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2) does exist.