Let $\Sigma_g$ be a surface of genus $g$. The *Heawood Conjecture* gives a closed formula in one variable, $\chi$ (the Euler characterstic of $\Sigma_g$), for the minimal number of of colors needed to color any graph drawn on $\Sigma_g$.

**Ringel-Youngs, 1968**: The Heawood Conjecture is true, except for $g=0$ (where we don't know the answer - this would be the *Four-Color Conjecture*) and the Klein bottle (where a graph was shown to be colorable with fewer than predicted by the formula).

Of course, the Four-Color Conjecture was eventually proven (not by Ringel or Youngs, although I think they were both involved to some extent), leaving the Klein bottle as the odd-man out.

(a) What is the "reason" for the Klein bottle's exceptionality here? (b) Does the answer to part (a) manifest in any other way - meaning, are there other theorems, patterns, etc to which the Klein bottle is an exception that can be traced back to (a)?