Maximal knotless graphs

Abstract

A graph is maximal knotless if it is edge maximal for the property of knotless embedding in $R^3$. We show that such a graph has at least $\frac{7}{4}|V|$ edges, and construct an infinite family of maximal knotless graphs with $|E|<\frac{5}{2}|V|$. With the exception of $|E|=22$, we show that for any $|E|\ge 20$ there exists a maximal knotless graph of size $|E|$. We classify the maximal knotless graphs through nine vertices and 20 edges. We determine which of these maxnik graphs are the clique sum of smaller graphs and construct an infinite family of maxnik graphs that are not clique sums.

14 pages, 3 figures v2: substantial revisions. Improved Theorem 4.5. Added Section 5

References in corpus (1)

• Intrinsic knotting and linking of almost complete graphs

Cited by in corpus (1)

• Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant