Lattice stick number of spatial graphs

TL;DR

This paper introduces an upper bound for the lattice stick number of spatial graphs, extending the concept from knots to graphs with vertices of degree up to six, based on their crossing number and structural properties.

Contribution

It provides a new upper bound formula for the lattice stick number of spatial graphs, generalizing previous knot-based results to more complex graph structures.

Findings

Derived an explicit upper bound for $s_L(G)$ in terms of crossing number and graph parameters.

Extended lattice stick number concepts from knots to spatial graphs with vertices of degree up to six.

Presented a formula connecting lattice stick number with crossing number, edges, vertices, and components.

Abstract

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number $s_{L}(G)$ of spatial graphs $G$ with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number $c(G)$ $$ s_{L}(G) \leq 3c(G)+6e-4v-2s+3b+k, $$ where $G$ has $e$ edges, $v$ vertices, $s$ cut-components, $b$ bouquet cut-components, and $k$ knot components.