2024/05/17 by Edgar A. Bering, Bennett Haffner, Bering, Edgar A. +5
Computer Science · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2405.10932
openalex publication_date 2024/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The sphere graph of Mr, a connect sum of r copies of S1× S2 was introduced by Hatcher as an analog of the curve graph of a surface to study the outer automorphism group of a free group Fr. 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 Mr. 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.