Stick number of non-paneled knotless spatial graphs

Abstract

We show that the minimum number of sticks required to construct a non-paneled knotless embedding of $K_4$ is 9 and of $K_5$ is 12 or 13. We use our results about $K_4$ to show that the probability that a random linear embedding of $K_{3,3}$ in a cube is in the form of a Möbius ladder is $0.97380\pm 0.00003$, and offer this as a possible explanation for why $K_{3,3}$ subgraphs of metalloproteins occur primarily in this form.

16 pages, 19 figures Corrected minimum number of sticks for $K_4$ in abstract to 9, reflecting the stated result in Theorem 2.6

Cited by in corpus (1)

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