Apr '26 Notes

This is my fourth set of monthly notes for this year where I write down interesting facts and ideas I have explored during my spare time. There were three things in particular that occupied my leisure time this month. First, I managed to learn the proof of Tutte's famous theorem that any \( s \)-arc-transitive finite cubic graph must satisfy \( s \le 5. \) I learnt the proof from Norman Biggs's book Algebraic Graph Theory. The original proof appears in Tutte's 1947 paper 'A family of cubical graphs' (DOI). Biggs's presentation differs considerably from Tutte's original argument and relies heavily on the properties of stabiliser sequences of arcs. I should say that Biggs's proof, while complete, is extremely condensed. The proof reads more like a high-level outline that moves rapidly from one main result to the next without sufficiently explaining the intermediate steps. As a result, it took considerable effort to work out the proofs of the intermediate results. Biggs presents the…