The freeness Index of a graph

TL;DR

This paper introduces the freeness index, a new graph invariant measuring how graphs can be embedded in 3-spheres with simple complements, and relates it to embedding problems and the orientable cycle double cover conjecture.

Contribution

It defines the freeness index, explores its properties, and connects it to embedding questions and the orientable cycle double cover conjecture.

Findings

Freeness index relates to embedding graphs with simple complements.

Cubic graphs satisfying the orientable double cycle cover conjecture have freeness index at least two.

Provides new insights into graph embedding and topological graph theory.

Abstract

We define a new integer invariant of a finite graph G, the freeness index, that measures the extent to which G can be embedded in the 3-sphere so that it and its subgraphs have ``simple" complements, i.e., complements which are homeomorphic to a connect-sum of handlebodies. We relate the freeness index to questions of embedding graphs into surfaces, in particular to the orientable cycle double cover conjecture. We show that a cubic graph satisfying the orientable double cycle cover conjecture has freeness index at least two.