IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

The structural of maximal non-biconnected unit distance graphs in the plane

  • April 9, 2021
  • 2:30 p.m.

Abstract

The famous Hadwiger-Nelson problem asks for the smallest natural number $k$ so that every point of the plane can be $k$-colored in such a way that no two points at a unit distance receive the same color. This question can be asked of any metric space, so we investigate the set of points in a disk of radius $R$ about the origin. It turns out that understanding this chromatic number as $R$ varies can shed light on the original question, and it raises several interesting questions about graphs embeddable into the unit distance graph of the plane with various rigidity properties. Our main result is a complete description of the maximal non-biconnected unit distance graphs in the plane, and the proof uses certain Diophantine equations and lots of elementary geometry. Joint work with Michael Filaseta and Kaylee Weatherspoon.

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