An underground cable used in power systems must have specific critical properties to ensure reliable and efficient operation. Let's evaluate each option:

1. High magnetic saturation capability inside cable conductor arrangements only: This property is more relevant to the design of magnetic materials and transformers, not directly applicable to underground cable requirements. Cables do not require high magnetic saturation capabilities since they primarily focus on electrical insulation and mechanical strength.
2. Low melting point of conductor material during fault conditions: A low melting point is undesirable because it means the conductor material can easily melt under fault conditions, leading to failure and potential hazards. Conductors should instead have a higher melting point to withstand the heat generated during faults.
3. Variable conductor diameter during balanced load operating conditions: This is not practical nor feasible for cables. Conductors should have a consistent diameter to ensure uniform current distribution and avoid hotspots. Variability could lead to inefficiencies and increased risk of faults.
4. High insulation resistance and adequate mechanical strength properties: This option is correct. High insulation resistance ensures that current does not leak through the insulation, maintaining safety and efficiency. Adequate mechanical strength is essential to withstand physical stresses during installation and operation, preventing damage and prolonging the cable's life.

Thus, the most important requirement for an underground cable in power systems is having high insulation resistance and adequate mechanical strength properties. This ensures safety, reliability, and longevity under varied environmental conditions.