Generation and New Infinite Families of K2-hypohamiltonian Graphs

Alapfogalmak

The authors present an algorithm to generate all pairwise non-isomorphic K2-hypohamiltonian graphs, improving upon earlier computational results by introducing new bounding criteria specifically designed for these graphs.

Kivonat

The content discusses the generation of K2-hypohamiltonian graphs using an algorithm that preserves non-hamiltonicity, K2-hamiltonicity, and planarity. The authors provide proofs and results on the existence and properties of these graphs, extending previous research findings.
The algorithm efficiently generates K2-hypohamiltonian graphs by considering obstructions and constraints that ensure the preservation of key characteristics. Through detailed analysis and proofs, the authors establish the validity of their approach in creating infinite families of such graphs with specific properties.
Key concepts include amalgamation operations to combine smaller graphs while maintaining essential graph properties like non-hamiltonicity and planarity. The study extends prior research on hypohamiltonian graph generation to focus on K2-hypohamiltonian structures with improved computational efficiency.
Overall, the content provides valuable insights into the generation and characteristics of K2-hypohamiltonian graphs, offering a significant contribution to mathematical graph theory research.

Statisztikák

There exists a hamiltonian cycle h1 in G1 − u − v.
There is a hamiltonian cycle h2 in G2 −a2 −a′ 2 containing b2b′ 2.
C(h1, h2, b1b′ 2, b′ 1b2) is a hamiltonian cycle in G−u−v.

Idézetek

"The amalgamation operation preserves non-hamiltonicity."
"Efficiently generating K2-hypohamiltonian graphs through careful consideration of obstructions."
"Algorithm ensures preservation of key graph properties during generation process."

Főbb Kivonatok

Generation and New Infinite Families of $K_2$-hypohamiltonian Graphs

by Jan Goedgebe... : arxiv.org 02-29-2024

https://arxiv.org/pdf/2311.10593.pdf

Mélyebb kérdések

How does the concept of amalgamation contribute to understanding graph generation beyond this specific context

The concept of amalgamation in graph theory contributes to understanding graph generation beyond the specific context discussed in the provided text by offering a versatile method for creating new graphs from existing ones while preserving certain properties. In this case, the amalgamation operation was used to combine two non-hamiltonian K2-hypohamiltonian graphs into a new non-hamiltonian graph. This approach can be extended to explore various combinations of graphs with different characteristics, leading to the creation of diverse and complex graph structures.
Amalgamation provides a systematic way to merge graphs while ensuring that specific properties are maintained or altered as desired. By leveraging this technique, researchers can generate novel families of graphs with unique features and study their properties systematically. This not only enhances our understanding of graph theory but also opens up possibilities for exploring new classes of graphs with distinct structural attributes.

What counterarguments exist against using algorithms for generating complex graph structures like K2-hypohamiltonian graphs

Counterarguments against using algorithms for generating complex graph structures like K2-hypohamiltonian graphs may include concerns about computational complexity, resource-intensive computations, and potential limitations in scalability. Generating specialized types of graphs often requires sophisticated algorithms that may have high time and space complexity, making them challenging to implement efficiently for large-scale problems.
Additionally, there could be arguments regarding the practical utility or real-world applications of studying such specialized graph families. While theoretical investigations into unique graph structures contribute valuable insights into mathematical concepts and combinatorial optimization, the applicability of these findings in practical scenarios might be limited.
Moreover, critics might raise questions about the generalizability of results obtained from studying highly specific types of graphs. Focusing on niche areas within graph theory could potentially restrict broader exploration and understanding of more fundamental principles that apply across various types of networks and systems.

How can the principles applied in this study be extended to explore other types of specialized graph families

The principles applied in this study can be extended to explore other types of specialized graph families by adapting similar techniques tailored to those specific contexts. For instance:

Exploring Hypotraceable Graphs: The methodology used for generating K2-hypohamiltonian graphs can be modified to investigate bipartite hypotraceable or other variants where every vertex-deleted subgraph has certain path-related properties.

Investigating Hypercubic Graphs: Similar algorithms could be developed to generate hypercubic (n-cube) Hamiltonicity-preserving transformations or operations on hypercube-like structures.

Studying Planar Traceable Graphs: The approach employed for planar K2-hypohamiltonian graphs could inspire research on planar traceable or Eulerian/non-Eulerian path-specific configurations within planar networks.
By applying analogous methodologies with appropriate modifications based on the desired characteristics or constraints unique to each type of specialized family, researchers can uncover new insights into diverse classes of structured networks beyond just hypohamiltonicity aspects explored here.