Paths and cycles

Definitions.
- *Walk*: sequence of vertices $v_0,v_1,\ldots,v_k$ with $\{v_i,v_{i+1}\}\in E$.
- *Path*: walk with all vertices distinct.
- *Cycle*: closed walk $v_0=v_k$ with $v_1,\ldots,v_{k-1}$ distinct.

Example. In a triangle $1$-$2$-$3$-$1$, the sequence $1$-$2$-$3$-$1$-$2$ is a walk but not a path (vertex $1$ repeats).

Conclusion. Every path is a walk; the converse fails when vertices repeat.