Question

Give an example of a graph without an articulation point, a Hamiltonian cycle, or a 2-coloring

Answer 1

The graph below does not have an articulation point or a hamiltonian cycle and uses 3 colors for graph coloring.

Answer 2

 A-----------B 
|       |
 |             /  |  \
 |           /    |    \
 |         E    |      F 
 |          \     |     /
 |            \   |   /
|       |
 D----------C 

In this graph:

• It does not have an articulation point, as removing any single vertex does not disconnect the graph.
• It has no Hamiltonian cycle, as no cycle visits each vertex exactly once.
• It does not have a valid 2-coloring, it requires at least 3 colors( Red, Green, Blue) to avoid adjacent vertices having the same color- Represented in colored format in the graph.