Stay with me 👀, because this is going to like you.

I came up with an idea that feels so simple and obvious that I can’t help but wonder why I hadn’t thought of it earlier. Maybe difficult to implement 😭 but the idea it is good and simple.

I’m curious if anyone has explored something similar, or if there are known related results 🤔

Would love to hear your thoughts!

I’ve been working on the Four Color Theorem 🎨, specifically on reducing planar 3-regular graphs down to a minimal configuration and then back reconstructing them step by step using, when necessary, different color Kempe chain switches techniques.

The challenge is that while my reduction strategy works in most cases ✅, there are configurations where the reconstruction gets stuck ❌ and Kempe switches don’t resolve the conflict. Only by shuffling the order of face selection 🔀 I’ve always been able to eventually color the map.

Here’s the idea 🚀: instead of trying to design the perfect reduction strategy upfront, why not start from the opposite direction?

Since I can already color arbitrarily complex graphs (even if it requires trial, backtracking, and random Kempe switches 🎲), I can treat successful colorings as “ground truth”.

From there, I can analyze the sequence of reductions on an already colored graph and observe which choices preserve a valid path back to a full coloring 🔍

In other words, rather than guessing the best reduction strategy, I can learn it from examples of graphs that have already been four-colored.