If my employer files my tax information incorrectly, causing me to owe money when I file my taxes, should I have to pay it or should they? Va.

At the company where I work all the hiring paperwork is done by a manager on the computer. You dont actually fill any of it out yourself. The manager asks you questions and inputs it. After working there for a few months a coworker and I compared pay stubs and I noticed he had one more line than me under the taxes section. After closer inspection we determined they were not taking federal taxes out of my checks, which is not the option I chose when I got hired. I spoke with my manager about and they said we would get it fixed. That was a week ago, I am still bugging them and hope to get it fixed today. But even if I do that still doesn''t change the last few months and when I file my taxes the government is going to want that money. But since I did not pick that option and my employer messed up not me I dont think I should have to pay. Do I have a leg to stand on? The state is Virginia, the state selecter will not let me scroll down to select it.