Step 1: Examine the evolution of library science education in India.Indian library science education boasts a significant past, with several prominent universities initiating relevant programs.\[\begin{array}{rl} \bullet & \text{Punjab University: Initiated the inaugural formal training in 1915 in Lahore.} \\ \bullet & \text{Madras University: S.R. Ranganathan commenced a certificate program in 1929, later upgraded to a diploma program in 1937.} \\ \bullet & \text{Banaras Hindu University (BHU): Launched its diploma program in 1941.} \\ \bullet & \text{University of Calcutta: Introduced its diploma program in 1945.} \\ \end{array}\]
Step 2: Correlate the year with the respective university.According to the established historical timeline, the University of Calcutta is identified as the university that began its one-year diploma program in 1945.