I have been reading about the four colour theorem and the fact that it is proved using a computer. My question is whether it is likely that we will ever achieve a proof without the use of a computer? If so, is there active research in finding this alternate proof?


It is possible that one day a proof comes out, but probably with different considerations than the actual one.

In 1969, Ringel and Youngs developed the concept of planar graphs on $g$-type surfaces, a surface with $g$ holes. For example a sphere has type $0$, a torus has type $1$, etc. They proved the following theorem (without a computer):

Given a planar graph $G$ on a $g$-type surface with $g>0$ : $$ \chi(G) \le \frac{7+\sqrt{1+48g}}{2} $$ This is quite interesting, as the bound with $g=0$ yields $\chi(G)\le 4$. In other words, if someone was able to generalize the theorem for $g=0$ (in its current form, $g>0$), it would prove the 4 color theorem quite elegantly! Unfortunately after 47 years today no one has been able to do that.

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