Coloring Spheres in 3–Manifolds

Authors

• Edgar A. Bering, San Jose State UniversityFollow
• Bennett Haffner, San Jose State University
• Estephanie Ortiz, San Jose State University
• Olivia Sanchez, San Jose State University

Publication Date

1-1-2026

Document Type

Article

Publication Title

New York Journal of Mathematics

Volume

32

First Page

618

Last Page

626

Abstract

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.

Keywords

graph coloring, Kneser graph, sphere graph

Department

Mathematics and Statistics

Recommended Citation

Edgar A. Bering, Bennett Haffner, Estephanie Ortiz, and Olivia Sanchez. "Coloring Spheres in 3–Manifolds" New York Journal of Mathematics (2026): 618-626.