A renewal approach to prove the Four Color Theorem unplugged, Part III: Diamond routes, canal lines and $\Sigma$-adjustments

Abstract

This is the last part of three episodes to demonstrate a renewal approach for proving the Four Color Theorem without checking by a computer. The first and the second episodes have subtitles: ``RGB-tilings on maximal planar graphs'' and ``R/G/B Kempe chains in an extremum non-4-colorable MPG,'' where R/G/B stand for red, green and blue colors to paint on edges and an MPG stands for a maximal planar graph. We focus on an extremum non-4-colorable MPG $EP$ in the whole paper. In this part we introduce three tools based on RGB-tilings. They are diamond routes, normal and generalized canal lines or rings and $\Sigma$-adjustments. Using these tools, we show a major result of this paper: no four vertices of degree 5 form a diamond in any extremum $EP$.