Of Love and War (A sequel to "Happiness")

"The End Begins"

After Dumbledore's death, the four didn't come back to Hogwarts because they are going to finish whatever Dumbledore started. They vowed that they are going to destroy Voldemort. But first they need to find the horcruxes first and they must find the Sword of Gryffindor.

When Harry killed the Basilisk with sword of Gryffindor at the Chamber of Secrets. The blade is impregnated with Basilisk venom. So the sword of Gryffindor can destroy horcruxes.