We prove that the linear chromatic number of any $k\times k$ pseudogrid is $\Omega(k)$. 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.

翻译：我们证明任何 $k\times k$ progrid 的线性色体数是 $\ Omega(k) $。 Kun 等人( Algorithmica, 2021年)的论据是,这一结果使图的树深度以其线性色体数的函数在树上有一个更紧的上层框,并提供了进一步的证据支持其推测,即任何图的树性色体深度都以其线性色体数的线性函数为上层框。