Consider a planar graph where every vertex is incident to at least 3 edges, and assign to each edge a weight equal to the sum of the degrees of its endpoints.
The answer to Every simple planar graph with $\delta\geq 3$ has an adjacent pair with $deg(u)+deg(v)\leq 13$ shows that some edge has weight less than or equal to 13.
What is the smallest $n$ so that every such graph has an edge cover (a set of edges such that every vertex is incident to at least one edge in the set) with average weight at most $n$?
In the question Can the vertices of a planar graph of min degree 3 be covered with edges of average weight ( sum of degrees) at most 13? it was shown that $n = 13$ does not suffice; a planar graph is exhibited where every edge cover has average weight greater than 13.
Does $n = 14$ suffice? If not, what is the smallest weight for which the vertices of a planar graph can be covered? Or, can we cover almost all vertices?
I am also interested in other questions/reference material pertaining to "light subgraphs", i.e. subgraphs of low average edge weight.
For instance, is there a long path with low degree vertices on average? There seems to be some reasonable material on low degree triangles, but I could not find any info on light graphs with 4 or more vertices.