My roommate said she is leaving in 10 days, i.e. did not give a 30 day notice, and there was not written lease agreement. We are here in Alameda County in Northern California. Does she have to pay her share of next month's rent?

In California a tenant must give 30 days notice to terminate their month-to-month tenancy even if there is no lease agreement. However that 30 days can begin at any time in the month and runs for 30 days from that point. So if she gave you notice today then she is responsible for her share of the rent until February 20th, 2013.

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My roommate said she is leaving in 10 days

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Also, the notice should be in writing, not oral.