by Melissa, edited by others
(probably 6 Mounykhion) This is a festival involving a procession of unmarried Athenian girls to the Delphinion, the shrine on the bank of the Ilissos River near the Olympieion, where both Apollo and Artemis were worshiped. The girls carried the symbol of supplication, branches of the sacred Athenian olive bound up with white wool. Plutarch connects the day with Theseus, since he sailed on this day to Krete with his fellow-victims to meet the Minotaur, having dedicated a suppliant bough before his departure. While Theseus is connected with Apollo in legend, it’s more likely that the Delphinia was devoted to Artemis since the 6th of the month is Her usual day and maidens are associated with Artemis not Apollo. In any case, the young women were probably asking protection and favor from Artemis for the female sex (Parke, p. 137)
Sources:
Parke, H.W., Festivals of the Athenians, 1977
A sample Delphinia ritual may be found here.