Coloring spheres in 3--manifolds
Edgar A. Bering IV, Bennett Haffner, Estephanie Ortiz, Olivia, Sanchez

TL;DR
This paper investigates the chromatic number of the sphere graph associated with certain 3-manifolds, providing bounds that extend known results from surface curve graphs to 3-manifold topology.
Contribution
It establishes upper and lower bounds for the chromatic number of the sphere graph of connected sums of $S^1\times S^2$, extending the analogy from surface curve graphs to 3-manifolds.
Findings
Bounds for the chromatic number of the sphere graph of $M_r$
Extension of bounds to any orientable 3-manifold via prime decomposition
Connection between sphere graph properties and 3-manifold topology
Abstract
The sphere graph of , a connect sum of copies of was introduced by Hatcher as an analog of the curve graph of a surface to study the outer automorphism group of a free group . Bestvina, Bromberg, and Fujiwara proved that the chromatic number of the curve graph is finite; bounds were subsequently improved by Gaster, Greene, and Vlamis. Motivated by the analogy, we provide upper and lower bounds for the chromatic number of the sphere graph of . As a corollary to the prime decomposition of 3-manifolds, this gives bounds on the chromatic number of the sphere graph for any orientable 3-manifold.
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Taxonomy
TopicsComputational Geometry and Mesh Generation
