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It is actually 0.00001 for nodes to relay your transaction.
No! As the owner of a node thats merely relaying transaction I can assure you that there is no such thing as a relay fee.
If you want a free transaction, leave the inputs there for a while. If the priority is high enough and size is small enough, you can send a no fee transaction.
This is very bad advice considering the inputs used. E.g. one of the inputs was 0.0000864 BTC which would take ~11k days to get a reasonable high priority in order to be send without fee.
[1] see this thread for more:
https://bitcointalk.org/index.php?topic=1044399.0-snip-
1) I was referring to the minimum TX fee for Bitcoin Core (V 0.10) nodes to relay the transaction. Sorry for mixing it up.
2) I was referring to OnkelPaul's advice of the 0.01BTC withdrawal thershold.