Expansion of gap-planar graphs

TL;DR

The paper investigates the expansion properties of $k$-gap-planar graphs and related drawing classes. By introducing the broader notion of $k$-gap-cover-planar graphs and analyzing $r$-shallow minors and topological minors, it derives a linear expansion bound $\nabla_r(G) \le 18(k+1)(r+1)$ for all $r\ge0$, and an improved bound $\widetilde{\nabla}_r(G) \le 8\sqrt{(2r+1)(k+1)}$ for shallow topological minors. These results extend to $(g,k)$-gap-planar and $(g,k)$-gap-cover-planar graphs on surfaces, with corresponding density, treewidth, and minor-related bounds, highlighting a robust sparsity framework across topologies. The authors further connect expansion bounds to colouring, obtaining degeneracy and acyclic chromatic bounds that depend polynomially on $k$ and the genus, and discuss open problems that aim to refine the structural understanding of these beyond-planar graph classes.

Abstract

A graph is $k$-gap-planar if it has a drawing in the plane such that every crossing can be charged to one of the two edges involved so that at most $k$ crossings are charged to each edge. We show this class of graphs has linear expansion. In particular, every $r$-shallow minor of a $k$-gap-planar graph has density $O(rk)$. Several extensions of this result are proved: for topological minors, for $k$-cover-planar graphs, for $k$-gap-cover-planar graphs, and for drawings on any surface. Application to graph colouring are presented.

Figures (3)

• Figure 1: 1-planar drawing of $K_6$
• Figure 2: A 1-gap-cover-planar drawing that is neither $O(1)$-gap-planar nor $O(1)$-cover planar.
• Figure 3: Contracting the edge $vw$ in the proof of \ref{['MinorClosed']}.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5: ER18
• Proposition 6
• Lemma 7
• proof