Wheel Classes in Kontsevich Graph Complex and Merkulov's Low-Valence Conjecture

Assar Andersson

paper

Wheel Classes in Kontsevich Graph Complex and Merkulov's Low-Valence Conjecture

arXiv:2604.11327

Abstract

We show that the wheel classes in the Kontsevich graph complex $G C_{d}$ admit representatives supported on graphs with only $3$ - and $4$ -valent vertices. This verifies that Merkulov's low-valence conjecture holds for the wheel classes. More precisely, for every $m \geq 2$ , we prove that the wheel graph $W_{2 m + 1}$ is homologous to an explicit linear combination of $2^{m - 2}$ graphs, each having only $3$ - and $4$ -valent vertices.