my input voltage on ULN2003 is greater than 2.9 volt i.e about 4.9 volts, then also the ULN2003 is not switching the relay.You don't understand how the ULN2003 works. Each channel is a darlington transistor, so the chip needs ground to operate.
If your not driving a relay (e.g. a LED), COM does not have to be connected. COM is the at the relay supply voltage and is the "other half" of all of the built in suppression diodes. If the relays were far away from the ULN chip, I MIGHT consider extra ones at the relay(s).
Datasheet: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwi10aHbmabQAhWF2YMKHfhpAMYQFgggMAA&url=http://www.ti.com/lit/ds/symlink/uln2003a.pdf&usg=AFQjCNHUdv8pRaS3Y-KFFvAIM8_JaWmeew&sig2=vng5jOPE_9sNsOAx7yhZew See section 5 for the functions of pin 9 and 10.
You will need to look at figures 7 & 8 http://www.ti.com/lit/ds/symlink/uln2003a.pdf?#page=9 along with the coil current to figure out how much voltage the driver will drop and what input current is required. You also have to make sure this exceed the pull-in voltage for the relay.
The input voltage typically has to be greater than 2.9 V to turn the driver on.