Why can't mesh analysis be used for non-planar circuits?
My old Uni copy of Network Analysis (Van Valkenburg) goes on for a chapter or three building the mathematical (topological) background, touching on Euler's solution of the Konigsberg bridge problem in 1735 and Kirchhoff in 1947 and Listing. 

What they call the "window pane" method allows the essential meshes to be identified by inspection- loops without internal loops, and essential branches are branches that don't cross other branches. 

If it's not planar, you can't draw it as so, so they suggest you should employ what they call the "tree and chord set" method to analyze the circuit. 

You could probably get a rigorous mathematical proof based on graph theory, but not from moi.