There are some popular theories flying around the internet about the possibility that Dumbledore may have accidentally made a horcrux and that the object containing the fragment was either Fawkes the Phoenix, The Elder Wand, or something else. Could Dumbledore have made a horcrux?
According to JK Rowling, probably not.
It turns out this theory reached the author herself and she responded on twitter.
ROWLING: The idea that anyone believes this is strangely upsetting to me.
Since the idea of her fans entertaining such thoughts is upsetting to her, we can take that to mean that at no point did Dumbledore make any sort of horcrux.
You have to commit murder to create a Horcrux. And while Dumbledore may have killed, there's no evidence that he's murdered anyone.
‘By an act of evil –the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: he would encase the torn portion –’
Half-Blood Prince | Horcruxes