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

Abstract

Interval graphs and proper interval graphs are well known graph classes, for which several generalizations have been proposed in the literature. In this work, we study the (proper) thinness, and several variations, for the classes of cographs, crowns graphs and grid graphs.We provide the exact values for several variants of thinness (proper, independent, complete, precedence, and combinations of them) for the crown graphs $CR_n$. For cographs, we prove that the precedence thinness can be determined in polynomial time. We also improve known bounds for the thinness of $n imes n$ grids $GR_n$ and $m imes n$ grids $GR_{m,n}$, proving that $left lceil rac{n-1}{3} ight ceil leq thin(GR_n) leq left lceil rac{n+1}{2} ight ceil$. Regarding the precedence thinness, we prove that $prec-thin(GR_{n,2}) = left lceil rac{n+1}{2} ight ceil$ and that $left lceil rac{n-1}{3} ight ceil left lceilrac{n-1}{2} ight ceil + 1 leq prec-thin(GR_n) leq left lceilrac{n-1}{2} ight ceil^2+1$. As applications, we show that the $k$-coloring problem is NP-complete for precedence $2$-thin graphs and for proper $2$-thin graphs, when $k$ is part of the input. On the positive side, it is polynomially solvable for precedence proper $2$-thin graphs, given the order and partition.