On asymptotic packing of convex geometric and ordered graphs

TL;DR

This paper establishes conditions under which convex geometric and ordered graphs can be packed into complete graphs, showing that graphs with certain chromatic numbers are always packable and identifying classes with many long edges that are also packable.

Contribution

It proves that convex geometric graphs with cyclic chromatic number at most 4 and ordered graphs with interval chromatic number at most 3 are always packable, extending understanding of graph packing.

Findings

Convex geometric graphs with cyclic chromatic number ≤ 4 are packable.

Ordered graphs with interval chromatic number ≤ 3 are packable.

Graphs with many long edges are also shown to be packable.

Abstract

A convex geometric graph $G$ is said to be packable if there exist edge-disjoint copies of $G$ in the complete convex geometric graph $K_n$ covering all but $o(n^2)$ edges. We prove that every convex geometric graph with cyclic chromatic number at most $4$ is packable. With a similar definition of packability for ordered graphs, we prove that every ordered graph with interval chromatic number at most $3$ is packable. Arguments based on the average length of edges imply these results are best possible. We also identify a class of convex geometric graphs that are packable due to having many "long" edges.