Equitable coloring of graphs beyond planarity
Weichan Liu
TL;DR
The paper investigates equitable coloring for graphs beyond planarity, focusing on IC-planar and NIC-planar graphs. It reduces the previously known degree thresholds for equitable $Δ(G)$-colorings from $Δ\ge12$ to $Δ\ge10$ for IC-planar graphs and from $Δ\ge13$ to $Δ\ge11$ for NIC-planar graphs by combining edge-based induction, 6-degeneracy, and a witness-switching method on a color-class directed graph $\mathcal{D}$. The technique analyzes accessibility/terminality of color classes, leverages strong components, and conducts a detailed case analysis to prove existence of equitable $r$-colorings for all $r\ge Δ(G)$ with the stated thresholds. Furthermore, the results extend to broader graph families defined by edge-density constraints, enriching the landscape of equitable coloring beyond planarity and illustrating how structural graph properties impact colorability. These insights open avenues for tightening thresholds and applying similar methods to other nonplanar graph classes.
Abstract
An equitable coloring of a graph is a proper coloring where the sizes of any two different color classes do not differ by more than one. A graph is IC-planar if it can be drawn in the plane so that no two crossed edges have a common endpoint, and is NIC-planar graphs if it can be embedded in the plane in such a way that no two pairs of crossed edges share two endpoints. Zhang proved that every IC-planar graph with maximum degree $Δ\geq 12$ and every NIC-planar graph with maximum degree $Δ\geq 13$ have equitable $Δ$-colorings. In this paper, we reduce the threshold from 12 to 10 for IC-planar graphs and from 13 to 11 for NIC-planar graphs.