The bone of contention between the Vijayanagar and Bahmani rulers was the Raichur Doab region.

Explanation:
1. Raichur Doab: The Raichur Doab is a region located between the Krishna and Tungabhadra rivers. It is an area of fertile land with rich agricultural resources. The region was strategically important as it served as a trade route between the Deccan plateau and the southern regions.

2. Vijayanagar Empire: The Vijayanagar Empire was a powerful Hindu kingdom that ruled over South India from the 14th to the 17th century. The empire was founded by Harihara and Bukka, who were appointed as governors of the Kampili kingdom by the Delhi Sultanate. Vijayanagar grew into a major power, controlling vast territories and becoming a center of art, culture, and trade.

3. Bahmani Sultanate: The Bahmani Sultanate was a Muslim kingdom that was established in the Deccan region in the 14th century. It was founded by Alauddin Hasan Bahman Shah, who revolted against the Delhi Sultanate. The Bahmani Sultanate was known for its patronage of Persian culture and architecture.

4. Conflict over Raichur Doab: The Raichur Doab region became a bone of contention between the Vijayanagar and Bahmani rulers due to its strategic importance and rich resources. Both empires sought to control this region to expand their territories and secure their trade routes.

5. Vijayanagar-Bahmani Wars: The Vijayanagar and Bahmani rulers engaged in a series of wars known as the Vijayanagar-Bahmani Wars. These wars were fought primarily over the control of the Raichur Doab region. The Vijayanagar rulers were initially successful in repelling the Bahmani attacks and maintaining their control over the region.

6. Battle of Talikota: However, the Vijayanagar Empire faced a major defeat in the Battle of Talikota in 1565. The combined forces of the Deccan Sultanates, including the Bahmanis, defeated the Vijayanagar army. This battle resulted in the decline of the Vijayanagar Empire and the subsequent fragmentation of the Deccan region.

In conclusion, the Raichur Doab region, located between the Krishna and Tungabhadra rivers, was the bone of contention between the Vijayanagar and Bahmani rulers. Both empires sought to control this strategically important and resource-rich region, leading to a series of wars and ultimately the decline of the Vijayanagar Empire.