The Icy Dragon Sword (commonly abbreviated to "IDS"), like the Fiery Dragon Sword, is a popular sword that is prized for its style. It has a unique look, with its frosty blade (only two twisted "filaments" as a solid core) coming out of the dragon's mouth that acts as the hilt. It only appears with a gold rarity.
Dye only affects the sword's hilt- it does not change the color of the flame. For a sword with a different color flame, consider the Fiery Dragon Sword. If you don't want an elemental mod, think about the Vampiric Dragon Sword.
The Icy Dragon Sword obtained from the Traveler cannot be dyed.