You can use the old REST API or the new Graph API. In any case, you must have publish_stream permission.
If you want to use the Graph API, you need to create a Post object and then send it to the graph feed URL for this profile
graph.facebook.com/PROFILE_ID/feed
You need to set the source property in the Post object for the youTube URL for the video.
http://developers.facebook.com/docs/reference/api/post/
If you want to use the old REST API, the process is similar, but you send a message
api-video.facebook.com/restserver.php
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