Could it be the ”House of Night” series by P. C. and Kristin Cast?
The front cover of the first book (Marked) has hot pink colours and tattoos play a big part in the storyline. The main character is a teenage vampire. In the story, mature vampires get elaborate tattoos from the goddess Nyx. The main character Zoey Redbird usually gets some sort of new tattoo at the end of each book as a result of some heroic or transformative deed. The book covers often depict the tattoos.
The young, fledgling vampires have to be near adult vampires or they get sick. They attend the boarding school House of Night where all the teachers are vampires. At one point, a small group of fledgling vampires leave the school and live in some tunnels. There is also a story arch whereby some of the pupils become "red" vampires (their tattoos are red and they die before maturing, i.e. they’re undead vampires). The red vampires tend to get sick before they die (then get evil for a while before they get better).
The "government" aspect could come from various aspects of the story depending on how far through the series you’ve read.