A few drawings to (hopefully) try to explain the relationship of relay 37KC and the fuel solenoid for engine operations on the A300-600. This example is for engine #1. Engine #2 uses 38KC.
One thing to keep in mind is that the solenoid acts similar to a latching relay. It will maintain its current condition until a command (in this case 28VDC) to change state is applied to one of its coils. In other words, if its closed it will remain closed until 28VDC is sent to the open coil and it will remain open until that voltage is applied to the closed coil.
I'm going to have to guess that the engineers did not want power applied to the fuel solenoid on a full time basis. Either there was a heat issue or longevity is increased without voltage always present.
Relay 37KC is basically "on" 99.9% of its life and for anyone who has brushed up against a A300 relay rack...... it's one of the ones that burns you.
Fuel Solenoid and Pressure Switch
Fuel solenoid end view and side view showing an extremely fine mesh.
A300-600 ATA 76-12 Engine Start Solenoid