🤖 AI Summary
This paper investigates the $chi$-binding function—i.e., the optimal upper bound on the chromatic number $chi(G)$ in terms of the clique number $omega(G)$—for intersection graphs of line segments in the plane admitting at most $d$ distinct slopes ($d$-DIR graphs). Employing combinatorial graph theory, geometric constructions, and extremal analysis, we determine the exact $chi$-binding function for this class: $chi(G) leq domega(G)$ if $omega(G)$ is even, and $chi(G) leq d(omega(G)-1)+1$ if $omega(G)$ is odd. We construct extremal graphs achieving these bounds, partially confirming a conjecture by Bhattacharya et al. Our result unifies and generalizes the classical theorem of Kostochka and Nešetřil for $d = 2$, thereby providing a complete characterization of the chromatic behavior of $d$-DIR graphs.
📝 Abstract
Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $omega$ that the chromatic number $chi(G)$ of $G$ is at most $domega$. We show for every even value of $omega$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvov{r}'ak and Noorizadeh. Furthermore, we show that the $chi$-binding function of $d$-DIR is $omega mapsto domega$ for $omega$ even and $omega mapsto d(omega-1)+1$ for $omega$ odd. This extends an earlier result by Kostochka and Nev{s}etv{r}il, which treated the special case $d=2$.
