Beginn der Vierfarbenforschung (1947–1955)

A discrete Gauss-Bonnet type theorem

“…The expectation of the Poincaré-Hopf index i f (v) over natural probability spaces like the uniform counting measure on colorings is the Levitt curvature [50,24,35]. For K 4 -free graphs, it leads to 1 − f 0 (S(v))/2 + f 1 (S(v))/3 which for 2-manifolds simplifies with f 1 (S(v)) = f 0 (S(v)) to 1 − f 0 (S(v))/6, a curvature which goes back to Victor Eberhard [9] and was used in graph coloring arguments already at the time of Birkhoff and heavily for the 4 color theorem program [4], as Gauss Bonnet χ(G) = 2 implies that there must be some vertices of degree 4 or 5, points of positive curvature.…”