Math Problem Statement

is this correct?

For each of the following, does there exist a graph on 5 vertices which has that list as the list of vertex degrees? (Choose all where a graph would exist.)

Solution

To determine if a list of vertex degrees can form a graph, we use the Handshaking Lemma which states that the sum of all vertex degrees must be even, because it is equal to twice the number of edges.

Let's analyze each option:

  1. 1, 2, 3, 4, 5: The sum is 15, which is odd. So, a graph cannot exist.

  2. 1, 1, 1, 2, 2: The sum is 7, which is odd. So, a graph cannot exist.

  3. 0, 1, 2, 2, 3: The sum is 8, which is even. So, a graph can exist.

  4. 4, 4, 4, 4, 4: The sum is 20, which is even. So, a graph can exist.

  5. 1, 1, 2, 2, 2: The sum is 8, which is even. So, a graph can exist.

The selected options are correct.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Vertex Degrees
Handshaking Lemma

Formulas

Sum of vertex degrees = 2 × Number of edges

Theorems

Handshaking Lemma

Suitable Grade Level

Grades 9-12

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