Because you said you are a newbie in graphs, the problem can be solved even without that. Thinking about bipartite matching for this just helps visualizing the problem in a good manner, we want to classify all the vertices in two sets such that every vertex in first set connects only one unique vertex in other.

For doing it without matching, start from the source ( the first unvisited vertex). Get one of its neighbour (or adjacent node). Assume these two nodes make one edge where the two nodes have degree one. Find the next unvisited node and recur for this new vertex now. In the end if we are not left with any unvisited nodes, that states our matching in recursive calls are correct.

You may refer to this idea here.