2018/06/13 by Bonato, Anthony, Kinnersley, William B. · 1 citation
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.1806.05286
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 ζ(G). We settle a conjecture of \citenisse1 by providing an upper bound on the localization number as a function of the chromatic 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.