Question

(a) Construct a graph with six vertices and degree sequence 1,1,2,2,3,3 . (b) Construct a graph with six vertices and degree sequence \(1.1 .3,3,3,3\). (c) Can you find at least two graphs with each of these degree sequences?

Short answer

Yes, at least two different graphs can be constructed for both sequences.

Step-by-step solution

Understanding Condition (a)

We need to construct a graph with six vertices and a degree sequence of \(1, 1, 2, 2, 3, 3\). This means there are six vertices where two vertices have degree 1, two have degree 2, and two have degree 3.

Construct Graph for Condition (a)

To satisfy the degree sequence \(1, 1, 2, 2, 3, 3\), connect the vertices in the following manner: Connect the two vertices of degree 1 to two different vertices of degree 3. Connect the remaining vertices of degree 3 to each other. Finally, connect the degree 2 vertices to each of the degree 3 vertices so that the degree condition holds.

Explore Additional Graph for Condition (a)

An alternative structure could be: Connect the two vertices of degree 1 to the same vertex of degree 3. Connect the degree 3 vertices with each other and distribute remaining connections as necessary to ensure all connections fulfill the conditions of degree sequence \(1, 1, 2, 2, 3, 3\).

Understanding Condition (b)

We need to construct a graph with six vertices and a degree sequence of \(1, 1, 3, 3, 3, 3\). This sequence describes six vertices with two of degree 1 and four of degree 3.

Construct Graph for Condition (b)

To fulfill \(1, 1, 3, 3, 3, 3\), connect each of the degree 1 vertices to a different degree 3 vertex. Then connect the remaining degree 3 vertices to each other ensuring none are left unconnected.

Explore Additional Graph for Condition (b)

For a second structure, connect the two degree 1 vertices to the same degree 3 vertex. Connect the degree 3 vertices so each vertex has three connections, ensuring no extra connections beyond what's required to maintain degree 3.

Key concepts

Degree Sequence

In graph theory, a **degree sequence** is a crucial concept that describes the connectivity of a graph. It is simply a list of vertex degrees presented in non-increasing order. The degree of a vertex is the number of edges connected to it. In essence, the degree sequence provides an overall profile of vertex connectivity within the graph.
For example, in a degree sequence like \(1, 1, 2, 2, 3, 3\), each number represents the degree of a vertex. Two vertices have a degree of 1, two have a degree of 2, and two have a degree of 3. This sequence helps in understanding how vertices in a graph are interconnected.

• Helps determine possible configurations of a graph's structure.
• Guides in graph construction by specifying connectivity needed.
• Useful in identifying isomorphic graphs (graphs that look different but are structurally identical).

By examining a degree sequence, we can predict and construct various valid graph layouts that adhere to the specified conditions.

Graph Construction

Creating a graph from a degree sequence involves connecting vertices to achieve the specified vertex degrees. The degree sequence serves as a blueprint that dictates how vertices need to be connected.
For example, for the degree sequence of \(1, 1, 2, 2, 3, 3\), we start by fulfilling the need for two vertices having a degree of 1 by linking them to two different vertices of degree 3.
After ensuring the vertices of degree 1 are properly connected, we continue with vertices of degree 2 and 3 by establishing connections among them as needed. Ensuring that each connection respects the demands of the degree sequence is key.
Graph construction from a degree sequence involves:

• Connecting vertices incrementally while verifying the degree conditions are met.
• Checking that all specified vertex degrees are satisfied without any excess edges.
• Exploring alternative graphs that can be formed from the same degree sequence by altering the arrangement of connections.

This process requires patience and sometimes creativity, especially when multiple configurations might fulfill the same degree conditions.

Graph Degrees

The **degree** of a vertex in a graph is foundational to understanding graph structures. It refers to the number of edges incident to a vertex, or equivalently, the number of connections a vertex has. In directed graphs, one might consider both in-degrees and out-degrees, but for undirected graphs, the degree concept remains straightforward.
The concept of vertex degrees helps in multiple ways such as:

• Identifying the spread of connections in a graph which can indicate critical vertices.
• Determining graph connectivity properties like whether the graph is well-connected or sparse.
• Enabling the application of algorithms that rely on degree information, like those determining Eulerian paths, cycles, or coloring complexities.

Calculating the degree of each vertex is a stepping stone in graph theory, as it aids in mapping out the entirety of the network's interconnections. Whether designing new graphs or understanding existing ones, vertex degrees provide essential insights into graph dynamics.