Question

Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and eight edges.

Short answer

Degree sequences: (4, 4, 4, 2, 2) and (4, 4, 3, 3, 2).

Step-by-step solution

Determine Total Degree

The degree of a vertex in a graph is the number of edges connected to it. For any graph, the sum of the degrees of all vertices is equal to twice the number of edges due to each edge being counted at both endpoints. Here, we have 8 edges, so the total degree is calculated as follows: \[ \text{Total degree} = 2 \times 8 = 16 \]

Consider Vertex Degree Constraints

Since the graph has 5 vertices and there are no isolated vertices, each vertex has a degree of at least 1. The degree of any vertex cannot exceed 4 because there are only 4 other vertices to connect to in a 5-vertex graph. Thus, the possible degrees for each vertex range from 1 to 4.

Identify Degree Sequences that Sum to Total Degree

Our goal is to find degree sequences (tuples of degrees) for the 5 vertices that sum to 16 based on Step 1 and meet the criteria from Step 2. Each degree sequence must consist of the numbers between 1 and 4, and their sum must equal 16.

Validate Potential Degree Sequences

Enumerate potential sequences and validate them: - (4, 4, 4, 2, 2): Each vertex's degree is valid given the constraints, and the total is 16. - (4, 4, 3, 3, 2): Meets the conditions, summing to 16. - No other useful permutation sums to 16 under these constraints without exceeding a vertex's degree limit or not meeting the total degree. Thus, these sequences are valid.

Key concepts

Degree Sequence

In the realm of graph theory, a degree sequence refers to a list or sequence of degrees of the vertices within a graph structure. The degree of a vertex is defined as the count of edges connected to it. For instance, if a vertex in a graph is connected to three other vertices, it has a degree of 3. In this context, solving for possible degree sequences involves ensuring these degrees add up correctly to twice the number of edges in the graph. Why twice? Each edge contributes to the degree of two vertices—one at each of its endpoints. Thus, the sum of all vertex degrees in a graph is equivalent to twice the total number of edges.

So, when we are given a problem where a graph consists of 8 edges, knowing that the sum of the degree sequence must be 16 helps in narrowing down potential solutions. When generating possible sequences, each choice must be validated against this rule, ensuring the sequence satisfies this sum condition.

Vertices

Vertices are fundamental components in a graph and represent the points where edges meet. In simpler terms, they can be thought of as dots or nodes that can be connected by lines (known as edges). In a 5-vertex graph, like the one discussed in this particular exercise, each vertex must connect to other vertices in some manner if no isolated vertex is allowed.

When analysing vertices for degree sequences, it's crucial to remember that each vertex in a graph can only connect to every other vertex once, which naturally caps its degree. For the given problem, with five vertices, the maximum degree any single vertex can possess is 4, since this is the total number of possible connections (other vertices) it has in the graph.

Edges

Edges are the links or connections between vertices in a graph. They are essentially the lines that form relationships or paths between two vertices. Understanding edges is important because they play a critical role in determining the degree of each vertex. Each edge contributes to the degree of two vertices, one at each end.

In the problem discussed, the graph contains 8 edges. To find valid degree sequences, one must ensure that the chosen degrees for vertices correspond to these 8 connections. Since a degree sequence sums up to twice the number of edges, verifying this sum ensures that all edges are accounted for and properly distributed among the vertices. Logical distribution of edges allows valid and practical degree sequences to surface.

Isolated Vertex

An isolated vertex occurs in a graph when a vertex has no connections, meaning its degree is zero. In some problems, isolated vertices are a significant concern, possibly reducing connectivity in the graph. However, in this problem, we're specifically instructed that no isolated vertices are present.

This constraint simplifies our analysis by ensuring that the minimum degree every vertex can have is at least 1. This key condition shifts the focus to finding degree sequences where all vertices remain interconnected within the specified framework (5 vertices and 8 edges). Thus, ensuring every vertex is connected helps both in achieving realistic graphs and fulfills the no-isolated-vertex requirement.