Linear versus centred chromatic numbers

Abstract

We prove that the linear chromatic number of any k × k pseudogrid is Ω(k). By an argument of Kun et al (Algorithmica, 2021), this result gives a tighter upper bound on the treedepth of a graph as a function of its linear chromatic number and gives further evidence in support of their conjecture that the treedepth of any graph is upper bounded by a linear function of its linear chromatic number.