 Solution

True for finite connected simple graphs. average degree of a graph at most two implies $e\le v$. A spanning tree has $v-1$ edges, leaving at most one extra edge. Hence the graph is a tree or unicyclic. trees are planar, and a unicyclic graph is a cycle with trees attached to its vertices, which can also be embedded in the plane. Connectedness is essential to this argument: a disconnected nonplanar component can be diluted by isolated vertices.