Coloring graphs of homologous spheres
Abstract
The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of Mr, a connect sum of r copies of S1 × S2 plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate 𝒮v(Mr), the subgraph of the sphere graph of Mr induced by spheres representing a single primitive homology class [v] ∈ H2(Mr;ℤ). We show that this graph is uniquely r−1 colorable. As a corollary, with r≥ 3 we obtain a nontrivial homomorphism from the Torelli subgroup IOr ≤ Out(Fr) to the symmetric group on r−1 symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of Mr with spheres deleted.
Related
- Bounds for B-coloring planar and outerplanar graphs
- COLORING SEMIMINIMAL CAYLEY GRAPHS
- Double-Partition Coloring problem: a unified approach for conflict-free coloring of hypergraphs and other coloring problems in graphs
- Coloring sparse random Cayley graphs
- Strong edge coloring of graphs with maximum degree $6$
BibTeX
@misc{bering2026coloring,
title = {{Coloring graphs of homologous spheres}},
author = {Edgar A. Bering IV and Samantha Ford and Aarav Ghai and Maxwell Lyon and Om Shah},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12347},
eprint = {2610.12347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12347},
}
RIS
TY - UNPB
TI - Coloring graphs of homologous spheres
AU - Edgar A. Bering IV
AU - Samantha Ford
AU - Aarav Ghai
AU - Maxwell Lyon
AU - Om Shah
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12347
UR - https://arxiv.org/abs/2610.12347
AB - The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of M_r, a connect sum of r copies of S_1 × S_2 plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate 𝒮_v(M_r), the subgraph of the sphere graph of M_r induced by spheres representing a single primitive homology class [v] ∈ H_2(M_r;ℤ). We show that this graph is uniquely r−1 colorable. As a corollary, with r≥ 3 we obtain a nontrivial homomorphism from the Torelli subgroup IO_r ≤ Out(F_r) to the symmetric group on r−1 symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of M_r with spheres deleted.
ER -