NP-hard problems on paths
A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. The problem is: given an edge-colored graph $G$ and an integer $k$, does $G$ have a rainbow matching with at least $k$ edges? This is known as rainbow matching problem, and its NP-complete even for properly edge-colored paths. The authors even note that prior to this result, no unweighted graph problem is known to be NP-hard for simple paths to the best of their knowledge.

See Le, Van Bang, and Florian Pfender. "Complexity results for rainbow matchings." Theoretical Computer Science (2013), or the arXiv version.