graph c++ theory

An undirected graph is called d-regular if, for every vertex v, (v) = d. For example, in a 3-regular graph, every vertex has three neighbors.

Suppose I have a d-regular graph and I want to find a path that contains d+ 1 (or more) vertices. It turns out this is easy to do: pick an arbitrary start vertex. Until I have a path with d + 1 vertices, pick an adjacent vertex to my current one that I have not yet visited. Add that edge to my growing path and set my current vertex to that one. Explain in 1-2 sentences why this will always produce a path with at least d+ 1 vertices. How do I know I wont get stuck at a vertex until it is at least the d + 1th vertex I visit? (GIVE A GOOD EXPLANATION FOR THIS IN PARTICULAR BECAUSE I DON'T UNDERSTAND THIS QUESTION