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Fabrication of Three-Dimensional Graphene-Based Polyhedrons via Origami-Like Self-Folding

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On Hamiltonian Decomposition Problem of 3-Arc Graphs.

Guangjun Xu¹, Qiang Sun², Zuosong Liang³

¹School of Mathematics, Zunyi Normal University, Zunyi, Guizhou, China.

Computational Intelligence and Neuroscience

|May 9, 2022

Summary

Every connected 3-arc graph contains multiple Hamilton cycles. For cubic graphs, the 3-arc graph, which is 4-regular, is proven to have a Hamiltonian decomposition.

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Area of Science:

Graph theory
Discrete mathematics

Background:

Introduces the concept of a 3-arc in graph theory.
Defines the 3-arc graph derived from an initial graph H.
Explains Hamiltonian cycles and Hamiltonian decomposition in graphs.

Purpose of the Study:

To investigate the properties of 3-arc graphs.
To determine if 3-arc graphs possess Hamiltonian cycles.
To explore Hamiltonian decomposition in the 3-arc graphs of cubic graphs.

Main Methods:

Graph-theoretic analysis of 3-arc graphs.
Utilizing the definition of 3-arcs and their relationship to paths.
Applying properties of Hamiltonian cycles and decompositions.

Main Results:

Proves that every connected 3-arc graph contains more than one Hamilton cycle.
Establishes that the 3-arc graph of a cubic graph is 4-regular.
Demonstrates that the 3-arc graph of a specific family of cubic graphs has a Hamiltonian decomposition.

Conclusions:

Connected 3-arc graphs are rich in Hamiltonian cycles.
The structure of 3-arc graphs, particularly for cubic graphs, supports Hamiltonian decomposition.
This research contributes to understanding the structural properties of derived graph structures.