Lombardi Drawings

A Lombardi drawing is a graph whose edges are circular arcs and whose vertices fulfil the property of perfect angular resolution. We prove that every subcubic graph has a Lombardi drawing. We use an old theorem from analyst Paul Koebe to proof this statement. We continue to prove Koebe’s theorem. First, we generalise circle packings to circle patterns in order to generalise its statement. Then, we prove the generalisation by using variational principles. Thereafter, Koebe’s theorem itself is a straight forward consequence. In conclusion, we given another geometric interpretation of the variational functional in the Appendix.