Mahanidhi Madan Gopal Swami: All the months of the lunar calendar are named after different forms of Narayana like Madhava, Govinda, Keshava and so on as in Vaisnava tilaka. Similarly, the month of Kartik is called "Damodara" which also refers to form of Vishnu. This fact is confirmed in Sanatana-siksha (Cc. 2.20.201), wherein Shri Chaitanya Mahaprabhu says:
asvine padmanabha, kartike damodara, radhadamodara anya vrajendra-kumara
"In the month of Äçvina, the predominating Deity is Padmanabha, and in Karttika it is Damodara. This Damodara [which the month is named after] is different from RadhaDamodara, the son of Nanda Maharaja in Våndavana
Nevertheless, most Vaishnavas understand that the "Damodara month" refers to that naughty boy, Krishna, who Mother Yashoda bound around the waist with a rope, which took place on Dipavali day in Kartik. During this Damodara month, if one manages to capture Krishna with the ropes of pure devotion, he will be released from the bondage of repeated birth and death.
Gaudiya Vaishnavas consider Radharani to be the presiding Deity for the month because Sri Srila Rupa Goswamipada (Utkalika-vallari) refers to Radharani as Kartik-devi, the goddess of Kartik. Unlike others who call it the "Damodara" month, Gaudiyas, using the following scriptural evidence, refer to Kartik as the "Radha-Damodara" month.
"One day in Kartik, Shyama arrived late for His rendezvous with Priyaji in Her kunja. In loving anger, Manini Radha glanced at Shyama with frowning eyebrows. Using some golden vines, Kunjeshvari then tied a rope around Shyama’s belly to punish Him for not showing up as promised. Madhava pleaded that He was late because Ma Yashoda had kept Him home for a festival. Seeing Her mistake, Radha quickly untied Her beloved Damodara." (Bhaviñya Purana, Uttara-khanòa) Radharané’s desire to bind Krishna with Her love is fulfilled in this pastime. Thus Vrajavasis adore this Damodara, bound by Devi’s love.
The Garga-samhita tells another wonderful lila of Radha-Damodara: "One day as children Radha and Gopala were walking together in Bhandiravana along the shimmering blue waters of Kalindi. The lovely couple rested for a time under the cooling shade of a Banyan tree. In an instant, Yogamaya manifested their robust, youthful forms. Suddenly from nowhere, Lord Brahma arrived. Then acting as the officiating priest, Brahmaji performed a mock marriage with all the demigods in attendance. During the wedding ceremony the sari and dhoti of Radha and Krishna were tied together as is the custom in Vedic weddings. After the ceremony the guests left, and Radha-Govinda again assumed their childhood forms and walked home."Since the Divine Couple were tied together at this time, some Vaishnavas consider this pastime a lila of Radha-Damodara.