Bounds on the localization number
Date of Original Version
We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written as ζ(G). We settle a conjecture of Bosek et al by providing an upper bound on the chromatic number as a function of the localization number. In particular, we show that every graph with ζ(G) ≤ k has degeneracy less than 3k and, consequently, satisfies χ(G) ≤ 3ζ(G). We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.
Journal of Graph Theory
Bonato, Anthony, and William B. Kinnersley. "Bounds on the localization number." Journal of Graph Theory 94, 4 (2020): 579-596. doi:10.1002/jgt.22546.