Denote by $N_c$ the non-orientable surface with $c$ pairwise disjoint crosscaps. So $N_1$ is the projective plane, $N_2$ is the Klein bottle, etc.
Then the "reason" of Klein bottle's exceptionality in Heawood conjecture is the following: for all $c \geq 3$, the complete graph $K_p$ embeds in $N_c$, where $p$ c$is the smallest integer which is at least$(p-3)(p-4)/6$. The case$c=2$is exceptional, in fact$K_7$does not embed in$N_2$. More details are here. This result on Klein bottle was proven by Franklin, who also showed that there actually exists in$N_2$a map with chromatic number$6$, obtained by embedding in the surface the so-called Franklin graph Post Undeleted by Francesco Polizzi 2 deleted 77 characters in body; edited body Denote by$N_c$the non-orientable surface with$c$pairwise disjoint crosscaps. So$N_1$is the projective plane,$N_2$is the Klein bottle, etc. Then the "reason" of Klein bottle's exceptionality in Heawood conjecture is the following: for all$c \geq 3$, the complete graph$K_p$embeds in$N_c$, where$p$is the smallest integer which is at least$(p-3)(p-4)/6$. The case$c=2$is exceptional, in fact$K_7$does not embed in$N_2$, and moreover every graph which can be embedded in the Klein bottle has chromatic number at most six. N_2$. More details are here.
This result on Klein bottle were was proven by Franklin, who also showed that there actually exists in $N_2$ a graph whith map with chromatic number $6$ embedded 6$, obtained by embedding in$N_2\$, the so called surface the so-called Franklin graph