Let $G=(V,E)$ be a graph where every "internal" vertex has degree 4, and every "external" vertex has degree $\le 3$. What can be said about this graph if it can be covered by a collection of edge-disjoint paths connecting internal vertices with external vertices? Did anybody study such problem? What are the possible obstructions? The question is related to amenability of the R.Thompson group $F$ (an approach of Victor Guba).
Clarification. For every internal vertex, one needs to choose a path connecting it to an external vertex, and all these paths (together) should be edge disjoint. You can view it as an evacuation plan. We need to evacuate all internal vertices using edges as bridges, but every edge can be used at most once.