Palettes of Dehn colorings for spatial graphs and the classification of vertex conditions

TL;DR

This paper explores Dehn colorings of spatial graphs, classifies vertex conditions, and demonstrates how these colorings can distinguish different spatial graphs, also extending the concept to generalized palettes for knot-theoretic structures.

Contribution

It introduces a classification of vertex conditions for Dehn colorings and extends palettes to knot-theoretic ternary-quasigroups, providing new tools for spatial graph analysis.

Findings

Spatial graphs can be distinguished by the number of Dehn colorings.

Classification of vertex conditions enhances understanding of spatial graph invariants.

Generalized palettes apply to knot-theoretic ternary-quasigroups.

Abstract

In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knot-theoretic ternary-quasigroups and region colorings of spatial graph diagrams.