The problem requires constructing graphs with specified properties:

(i) Graph with four vertices of degrees 1, 1, 3, and 3:

1. Label the vertices as $A, B, C, D$.
2. Connect $A$ and $B$ (both of degree 1).
3. Connect $C$ and $D$ with three edges (multigraph) to ensure they have degrees 3.

This creates the required graph with the degrees: 1, 1, 3, 3.

(ii) Graph with six vertices, each of degree 3:

1. Create a regular graph where each vertex is connected to exactly 3 others.
2. Label the vertices $A, B, C, D, E, F$.
3. Construct edges such as:
   • $A-B, A-C, A-D$
   • $B-C, B-E, B-F$
   • $C-D, C-E, C-F$
   • $D-E, D-F, E-F$.

This ensures each vertex has exactly degree 3. A 3-regular graph with 6 vertices is possible because the sum of degrees is even.

Would you like me to draw the graphs or explain the construction in more detail?

Follow-Up Questions:

1. What is a multigraph, and how does it differ from a simple graph?
2. How is the degree of a vertex defined, and why is it important in graph theory?
3. Can a graph with odd total degree exist? Why or why not?
4. What are examples of real-world networks that could be modeled by such graphs?
5. How do Eulerian graphs relate to the degree properties of vertices?

Tip:

Always verify if the degree sequence is graphical (i.e., can correspond to a graph) using algorithms like the Havel-Hakimi method.