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Interval graphs and proper interval graphs are well known graph classes, forwhich there have been proposed several generalizations in the literature. Inthis work, we study the (proper) $k$-thin graphs and its variations for theclasses of cographs, crown graphs and grid graphs.  We provide the exact values for several variants of thinness (proper,independent, complete, precedence, and combinations of them) for the crowngraphs $CR_n$. For cographs, we prove that the precedence thinness can bedetermined in polynomial time. We also improve known bounds for the thinness of$n \times n$ grids $GR_n$ and $m \times n$ grids $GR_{m,n}$, proving that$\left \lceil \frac{n-1}{3} \right \rceil \leq \mbox{thin}(GR_n) \leq \left\lceil \frac{n+1}{2} \right \rceil$. Regarding the precedence thinness, weprove that $\mbox{prec-thin}(GR_{n,2}) = \left \lceil \frac{n+1}{2} \right\rceil$ and that $\left \lceil \frac{n-1}{3} \right \rceil \left\lceil\frac{n-1}{2} \right \rceil + 1 \leq \mbox{prec-thin}(GR_n) \leq \left\lceil\frac{n-1}{2} \right \rceil^2+1$. As applications, we show that the$k$-coloring problem is NP-complete for precedence $2$-thin graphs and forproper $2$-thin graphs, when $k$ is part of the input. On the positive side, itis polynomially solvable for precedence proper 2-thin graphs, given the orderand partition.

Thinness and its variations on some graph families and coloring graphs of bounded thinness