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

TL;DR

This paper proves that wheel classes in the Kontsevich graph complex can be represented with graphs having only 3- and 4-valent vertices, confirming Merkulov's low-valence conjecture for these classes.

Contribution

It demonstrates that wheel classes in the Kontsevich graph complex admit representatives supported on graphs with only 3- and 4-valent vertices, verifying a specific conjecture.

Findings

Wheel classes are homologous to graphs with only 3- and 4-valent vertices.

Explicit linear combinations of graphs support the homology.

Verification of Merkulov's low-valence conjecture for wheel classes.

Abstract

We show that the wheel classes in the Kontsevich graph complex $GC_d$ 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 $m \ge 2$, we prove that the wheel graph $W_{2m+1}$ is homologous to an explicit linear combination of $2^{m-2}$ graphs, each having only $3$- and $4$-valent vertices.