Non-Homotopic Drawings of Multigraphs

Abstract

A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges (self-)intersects at most $k$ times. We prove that the number of edges in an $n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13 n (k + 1)}$, which is a big improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, $m$, in such a drawing is at most $2 \binom{2n}{k + 1}$ and the number of crossings is $Ω\bigl(\frac{m^{2 + 1/k}}{n^{1 + 1/k}}\bigr)$. For fixed $k$ these bounds are both best possible up to a constant multiplicative factor.

Figures (9)

• Figure 1: Non-homotopic edges (with no self-crossings) winding around the same vertex increasingly many times.
• Figure 2: Non-homotopic parallel edges.
• Figure 3: Example of the $n=5$ case: (a) drawing of $G$, (b) $F_{n-1}$ and (c) $F_n$, where $f_{1,1}$$f_{1,2}$ and $f_{1,3}$ are obtained by subdividing $f_1$.
• Figure 4: Building $F$.
• Figure 5: A face $F$ of length $3$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof : Proof of \ref{['NonHomotopic']}
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of \ref{['NonHomotopicxy']}