I wonder if it's "flapping" because something is causing the port to reset. The disconnection happens when the Ethernet PHY is forced to renegotiate the link (it takes 2-3 seconds).
This can happen if someone clicks "repair" on the interface - Windows brings the interface down and back up, which forces a renegotiation. It then does a new DHCP request to bring up the link.
Try setting your PC to use a static IP and see what happens.
What happens is if the lease is 1 hour, the PC will automatically do a lease renew when the lease is half expired. Thus at one hour, the PC will begin renewing the lease 1/2 hour in. When this happens, the DHCP server could very well send a DHCKNAK saying "renewal failed, drop your existing leases". Windows could very well reset the port (in case the network changed on it and it didn't notice - the usual cause of NAKs).
Usually caused by a subtle incompatibility between the DHCP server and client.
If static IPs work, use them (and make sure to exclude them from the DHCP range).