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.

If not, 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?

I am also interested in other questions/reference material pertaining to "light subgraphs", ( subgraphs of low degree)

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.

I have posted this question on Math.SE, but got no replies

everyedge has weight $5+5=10$.) $\endgroup$ – Noam D. Elkies Aug 15 '19 at 2:53averageweight for a cover of the wheel is $\sim 8$ for large wheels. A wheel with $n$ vertices on the rim can be covered by $\sim n/2+1$ edges, one of which has weight $n+3$, and the other have weight $6$. $\endgroup$ – M. Winter Aug 15 '19 at 7:34