Arc index of spatial graphs

TL;DR

This paper extends the concept of arc index from prime links to spatial graphs, establishing a tight upper bound based on crossing number, edges, and bouquet cut-components, thus generalizing and optimizing previous results.

Contribution

It introduces a new upper bound on the arc index for spatial graphs, generalizing prior work on prime links and proving the bound is optimal.

Findings

Upper bound on arc index:  c(G)+e+b

Bound is proven to be the lowest possible

Extension of arc presentation to spatial graphs

Abstract

Bae and Park found an upper bound on the arc index of prime links in terms of the minimal crossing number. In this paper, we extend the definition of the arc presentation to spatial graphs and find an upper bound on the arc index $\alpha (G)$ of any spatial graph $G$ as $$\alpha(G) \leq c(G)+e+b,$$ where $c(G)$ is the minimal crossing number of $G$, $e$ is the number of edges, and $b$ is the number of bouquet cut-components. This upper bound is lowest possible.