On asymptotic packing of convex geometric and ordered graphs

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.

Figures (5)

• Figure 1: All packable plane Hamiltonian cgg's.
• Figure 2: A cgg $H$; A weighted representation of $H$; The blowup $H[2]$.
• Figure 3: A sub-cgg $G'\subset H[2]$; Its weighted representation $W'\subset H$. (Edges of $H$ and $H[2]$ are not pictured here but in Figure \ref{['fig:weighted_and_blowup']}.)
• Figure 4: Configurations of $S_i$, $S'_i$, and $S"_i$. Numbers represent the edge-lengths.
• Figure 5: Configurations of $S_i$ and $S_{ij}$ for $k=3$. Numbers represent the edge-lengths.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Theorem 2.2: Rödl Nibble
• proof : Proof of Lemma \ref{['frac_pack_lemma']}
• Lemma 3.1
• proof : Proof of Theorem \ref{['main']}
• proof
• Lemma 3.3: Farkas' Lemma