innsikt - Graph Theory - # Graph Coloring

Open Problems from the 32nd Workshop on Cycles and Colourings

Q: What are the broader implications of finding (or not finding) a 5-edge-connected 5-regular Class 2 graph for other open problems in graph theory?

Answer: The existence of a 5-edge-connected 5-regular Class 2 graph is intimately linked to several prominent conjectures in graph theory. If such a graph is found, it would: Provide a counterexample to a potential strengthening of Thomassen's conjecture: Thomassen conjectured that highly edge-connected r-regular graphs (for sufficiently large r) should have two disjoint perfect matchings. A 5-edge-connected 5-regular Class 2 graph would be a counterexample for the case r=5, implying a lower bound on the potential value of r in a true version of the conjecture. Leave open the question of classifying all r-edge-connected r-regular Class 2 graphs: While such graphs are known for all other values of r, the case of r=5 remains elusive. Finding an example would complete this classification. If such a graph is proven not to exist, it would: Imply the Berge-Fulkerson Conjecture: This conjecture posits that every bridgeless cubic graph has a collection of six perfect matchings covering each edge exactly twice. The non-existence of the 5-edge-connected 5-regular Class 2 graph would provide a proof for this long-standing conjecture. Imply the 5-cycle double cover conjecture: This conjecture states that every bridgeless graph has a collection of five even subgraphs covering each edge exactly twice. Similar to the Berge-Fulkerson Conjecture, the non-existence result would lead to a proof of this conjecture. Support a potential strengthening of Thomassen's conjecture: The non-existence would suggest that 5 might be a sufficient value for r in Thomassen's conjecture, potentially leading to further investigations for a proof. Therefore, the existence or non-existence of a 5-edge-connected 5-regular Class 2 graph has significant implications for our understanding of edge-colorings, perfect matchings, and the structure of regular graphs.

Q: Could there be a connection between the difficulty in characterizing subcubic K3-free graphs of Class 2 and the unknown complexity of recognizing snarks?

Answer: Yes, there is a strong connection between the difficulty in characterizing subcubic K3-free graphs of Class 2 and the challenge of recognizing snarks. Snarks as a subclass: Snarks, by definition, are cubic graphs of Class 2. Since all snarks are also K3-free, the class of subcubic K3-free graphs of Class 2 contains all snarks. Inherited difficulty: The difficulty in characterizing snarks stems from their elusive structure and the lack of easily verifiable properties that distinguish them. This difficulty directly carries over to the broader class of subcubic K3-free graphs of Class 2. Any characterization of this broader class would necessarily need to account for the complexity of snarks. Subdivision as a complicating factor: The process of subdividing an edge of a cubic graph to obtain a subcubic graph can introduce additional complexity. Even if we had a complete characterization of snarks, determining whether a given subcubic K3-free graph of Class 2 can be obtained by subdividing a snark (and identifying which snark) could be a non-trivial task. In essence, the problem of characterizing subcubic K3-free graphs of Class 2 is at least as hard as characterizing snarks. Any progress in understanding and characterizing snarks would likely provide valuable insights into the broader class, but the subdivision aspect adds another layer of complexity that needs to be addressed.

Grunnleggende konsepter

This document presents a compilation of open problems in graph theory, specifically focusing on cycles and colorings, stemming from discussions at the 32nd Workshop on Cycles and Colourings held in Poprad, Slovakia.

Sammendrag

This research paper presents a collection of open problems in graph theory, specifically focusing on graph coloring, as discussed at the 32nd Workshop on Cycles and Colourings.

Bibliographic Information: Open problems of the 32nd Workshop on Cycles and Colourings (edited by Alfr´ed Onderko). arXiv:2411.10046v1 [math.CO] 15 Nov 2024

Research Objective: The paper aims to disseminate open problems related to cycles and colorings in graphs, fostering further research and advancements in the field.

Methodology: The paper compiles open problems presented and discussed by participants of the 32nd Workshop on Cycles and Colourings. Each problem is accompanied by a brief description, relevant background information, and references to existing literature.

Key Findings: The paper highlights several intriguing open problems, including:

Relaxations of conditions for crumby colorings in 3-connected cubic graphs.
Bounds and properties of list packing numbers in various graph classes.
The existence of 5-edge-connected 5-regular Class 2 graphs and their implications for perfect matchings.
Characterization of subcubic K3-free graphs of Class 2.
Algorithmic aspects of determining special vertex orderings related to anti-Ramsey problems.

Main Conclusions: The open problems presented in the paper represent significant challenges and research directions in graph theory, particularly in the areas of graph coloring, list coloring, and structural graph theory.

Significance: By disseminating these open problems, the paper aims to stimulate further research and collaboration within the graph theory community, potentially leading to new insights and breakthroughs in the field.

Limitations and Future Research: The paper focuses on a specific set of open problems discussed at a particular workshop. Further research is encouraged to explore these problems in greater depth and investigate related questions arising from the presented challenges.

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Open problems of the 32nd Workshop on Cycles and Colourings

by Jáno... klokken arxiv.org 11-18-2024

https://arxiv.org/pdf/2411.10046.pdf

Open problems of the 32nd Workshop on Cycles and Colourings

Dypere Spørsmål

What are the broader implications of finding (or not finding) a 5-edge-connected 5-regular Class 2 graph for other open problems in graph theory?

Answer:
The existence of a 5-edge-connected 5-regular Class 2 graph is intimately linked to several prominent conjectures in graph theory.
If such a graph is found, it would:

Provide a counterexample to a potential strengthening of Thomassen's conjecture:  Thomassen conjectured that highly edge-connected r-regular graphs (for sufficiently large r) should have two disjoint perfect matchings. A 5-edge-connected 5-regular Class 2 graph would be a counterexample for the case r=5, implying a lower bound on the potential value of r in a true version of the conjecture.
Leave open the question of classifying all r-edge-connected r-regular Class 2 graphs: While such graphs are known for all other values of r, the case of r=5 remains elusive. Finding an example would complete this classification.
If such a graph is proven not to exist, it would:

Imply the Berge-Fulkerson Conjecture: This conjecture posits that every bridgeless cubic graph has a collection of six perfect matchings covering each edge exactly twice. The non-existence of the 5-edge-connected 5-regular Class 2 graph would provide a proof for this long-standing conjecture.
Imply the 5-cycle double cover conjecture: This conjecture states that every bridgeless graph has a collection of five even subgraphs covering each edge exactly twice.  Similar to the Berge-Fulkerson Conjecture, the non-existence result would lead to a proof of this conjecture.
Support a potential strengthening of Thomassen's conjecture:  The non-existence would suggest that 5 might be a sufficient value for r in Thomassen's conjecture, potentially leading to further investigations for a proof.
Therefore, the existence or non-existence of a 5-edge-connected 5-regular Class 2 graph has significant implications for our understanding of edge-colorings, perfect matchings, and the structure of regular graphs.

Could there be a connection between the difficulty in characterizing subcubic K3-free graphs of Class 2 and the unknown complexity of recognizing snarks?

Answer:
Yes, there is a strong connection between the difficulty in characterizing subcubic K3-free graphs of Class 2 and the challenge of recognizing snarks.

Snarks as a subclass: Snarks, by definition, are cubic graphs of Class 2.  Since all snarks are also K3-free, the class of subcubic K3-free graphs of Class 2 contains all snarks.

Inherited difficulty: The difficulty in characterizing snarks stems from their elusive structure and the lack of easily verifiable properties that distinguish them. This difficulty directly carries over to the broader class of subcubic K3-free graphs of Class 2. Any characterization of this broader class would necessarily need to account for the complexity of snarks.

Subdivision as a complicating factor: The process of subdividing an edge of a cubic graph to obtain a subcubic graph can introduce additional complexity. Even if we had a complete characterization of snarks, determining whether a given subcubic K3-free graph of Class 2 can be obtained by subdividing a snark (and identifying which snark) could be a non-trivial task.
In essence, the problem of characterizing subcubic K3-free graphs of Class 2 is at least as hard as characterizing snarks.  Any progress in understanding and characterizing snarks would likely provide valuable insights into the broader class, but the subdivision aspect adds another layer of complexity that needs to be addressed.

How might the study of special vertex orderings in graphs contribute to advancements in areas beyond anti-Ramsey problems and odd-coloring?

Answer:
The study of special vertex orderings, like the Type-A and Type-B orderings described, has the potential to impact various areas beyond anti-Ramsey problems and odd-coloring. This is because these orderings capture fundamental structural properties of graphs that have implications for:

Algorithmic graph theory:

Efficient graph algorithms: The existence of specific orderings can lead to the development of more efficient algorithms for graph problems. For example, algorithms for graph coloring, matching, or finding independent sets can potentially be optimized by exploiting the properties of these orderings.
Graph decomposition:  Type-A and Type-B orderings could provide insights into decomposing graphs into simpler structures. This has applications in divide-and-conquer algorithms and understanding the structural composition of complex networks.

Graph representation and data structures:

Compact graph representations: Special orderings can lead to more compact ways of representing graphs, reducing storage space and improving the efficiency of data structures used to store and manipulate graphs.
Graph drawing and visualization:  Orderings can be used to develop algorithms for drawing graphs in a way that highlights specific structural properties, making them easier to visualize and analyze.

Network analysis and modeling:

Social network analysis:  In social networks, these orderings might reflect hierarchies or information flow patterns. Identifying such orderings can reveal influential individuals or communities within the network.
Biological networks:  In biological networks, such as protein-protein interaction networks, special orderings might correspond to biological pathways or functional modules.

Extremal graph theory:

Forbidden substructures: The existence or non-existence of specific orderings can be used to characterize classes of graphs based on forbidden substructures, leading to new results in extremal graph theory.
Overall, the study of special vertex orderings provides a powerful lens through which to analyze and understand graph structure. The insights gained from this study have the potential to advance various fields, ranging from theoretical computer science and discrete mathematics to applied areas like network science and bioinformatics.