Geochromatic Number when Crossings are Independent

A geometric graph, $\overline{G}$, is a graph drawn in the plane, with straight line edges and vertices in general position. A geometric homomorphism between two geometric graphs $\overline{G}$, $\overline{H}$ is a vertex map $f:\overline{G}\to\overline{H}$ that preserves vertex adjacency and edge crossings. The geochromatic number of $\overline{G}$, denoted $X(\overline{G})$, is the smallest integer $n$ so that there is a geometric homomorphism from $\overline{G}$ to some geometric realization of $K_n$. Recall that the chromatic number of an abstract graph $G$, denoted $\chi(G)$, is the smallest integer $n$ for which there is a graph homomorphism from $G$ to $K_n$. It is immediately clear that $\chi(G)\leq X(\overline{G})$. This paper establishes some upper bounds on $X(\overline{G})$ in terms of $\chi(G)$. For instance, if all crossings are at distance at least 1 from each other, then $X(\overline{G})\leq 3\chi(G)$. However, there are more precise results. If all crossing are at distance at least 2, then $X(\overline{G})\leq \chi(G)+2$. If all crossings are at distance at least 1, and there is a graph homomorphism $f: G \to K_n$ that maps no pair of edges that cross in $\overline{G}$ to the same edge in $K_n$, then $X(\overline{G})\leq 2n$. Finally, if $\chi(G)\in \{2,3\}$ and all crossings are at distance at least 1, then $X(\overline{G})\leq 2\chi(G)$.