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 $${\mathbb R}^2$$ 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 $$\omega $$ ω that the chromatic number $$\chi (G)$$ χ ( G ) of G is at most $$d\omega $$ d ω . We show for every even value of $$\omega $$ ω how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the $$\chi $$ χ -binding function of d-DIR is $$\omega \mapsto d\omega $$ ω ↦ d ω for $$\omega $$ ω even and $$\omega \mapsto d(\omega -1)+1$$ ω ↦ d ( ω - 1 ) + 1 for $$\omega $$ ω odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case $$d=2$$ d = 2 .

Abstract. A circular [Formula: see text]-coloring of a signed graph [Formula: see text] is an assignment [Formula: see text] of points of a circle [Formula: see text] of circumference [Formula: see text] to the vertices of [Formula: see text] such that for each positive edge [Formula: see text] of [Formula: see text] the distance of [Formula: see text] from [Formula: see text] is at least 1 and for each negative edge [Formula: see text] the distance of [Formula: see text] from the antipode of [Formula: see text] is at least 1. The circular chromatic number of [Formula: see text], denoted [Formula: see text], is the infimum of [Formula: see text] such that [Formula: see text] admits a circular [Formula: see text]-coloring. This notion was recently defined by Naserasr, Wang, and Zhu, who, among other results, proved that for any signed [Formula: see text]-degenerate simple graph [Formula: see text] we have [Formula: see text]. For [Formula: see text], examples of signed [Formula: see text]-degenerate simple graphs of circular chromatic number [Formula: see text] are provided. But for [Formula: see text] only examples of signed 2-degenerate simple graphs of circular chromatic number arbitrarily close to 4 are given, noting that these examples are also signed bipartite planar graphs. In this work we first observe the following restatement of the 4-color theorem: If [Formula: see text] is a signed bipartite planar simple graph where vertices of one part are all of degree 2, then [Formula: see text]. Motivated by this observation, we provide an improved upper bound of [Formula: see text] for the circular chromatic number of a signed 2-degenerate simple graph on [Formula: see text] vertices and an improved upper bound of [Formula: see text] for the circular chromatic number of a signed bipartite planar simple graph on [Formula: see text] vertices. We then show that each of the bounds is tight for any value of [Formula: see text].