 Solution

For a finite graph choose a simple path of maximum length. Every neighbor of its final vertex must already lie on the path, since otherwise that vertex could be appended. The final vertex has degree at least two, so besides its predecessor it has another neighbor earlier on the path. That edge together with the intervening path segment makes a cycle in a graph.

Finiteness is needed for this assertion in general: the two-sided infinite path has degree two at every vertex and no cycle in a graph. The customary finite-graph convention is used here and in the subsequent cube questions.