Chromatic number of intersection graphs of segments with two slopes
Sudatta Bhattacharya, Zdenĕk Dvořák, Fariba Noorizadeh · 2023
A \emph{$d$-dir} graph is an intersection graph of segments, where the segments have at most $d$ different slopes. It is easy to see that a $d$-dir graph with clique number $\omega$ has chromatic number at most $d\omega$. We study the chromatic number of $2$-dir graphs in more detail, proving that this upper bound is tight even in the fractional coloring setting.