insight - Scientific Computing - # Graph Theory

The Minimum Number of Edges in a 3-Connected Graph Where Every Neighborhood Contains a Cycle

Q: How does the minimum size change if we consider k-connected locally nonforesty graphs for k > 3?

Finding the minimum size of a k-connected locally nonforesty graph for k > 3 becomes more intricate. Here's why: Increased Complexity: Higher connectivity (k > 3) imposes stronger restrictions on the graph's structure. The techniques used in the paper for 3-connected graphs, like analyzing the neighborhoods of minimum degree vertices and using the degree-sum formula, might not directly generalize to higher connectivity. Dependence on Subgraph Structure: The requirement for each local subgraph to contain a cycle heavily influences the minimum size. As k increases, the local subgraphs need to be denser to ensure k-connectivity while still containing cycles. This interplay between connectivity and the cycle condition makes the problem significantly harder. Potential for New Constructions: The paper provides constructions for 3-connected locally nonforesty graphs. For k > 3, we might need entirely different construction techniques to achieve minimum size, potentially involving more complex base graphs and connection patterns. In summary, determining the minimum size for k > 3 is an open problem that would likely require new methods and a deeper understanding of the interplay between connectivity, local subgraph structure, and forbidden subgraphs.

Q: Could there be a relationship between the size of a locally nonforesty graph and its diameter or other graph invariants?

Yes, there could be relationships between the size of a locally nonforesty graph and other graph invariants like diameter: Diameter and Size: Locally nonforesty graphs with small diameters tend to have larger sizes. A small diameter enforces a certain density, as vertices need to be "close" to each other, leading to more edges. Conversely, larger diameters might allow for sparser constructions while still maintaining the locally nonforesty property. Other Invariants: Girth: The girth (length of the shortest cycle) of a locally nonforesty graph is related to its size. Smaller girth often implies a larger size, as more edges are needed to create short cycles within each local subgraph. Minimum Degree: As seen in the paper, the minimum degree plays a crucial role in bounding the size. Higher minimum degrees generally lead to larger sizes in locally nonforesty graphs. Independence Number: The independence number (size of the largest independent set) could be inversely related to the size. A large independent set might make it harder to ensure the locally nonforesty property without adding many edges. Exploring these relationships could involve: Finding bounds: Derive bounds on the size of locally nonforesty graphs based on their diameter, girth, or other invariants. Characterizing extremal graphs: Identify graphs that achieve equality in these bounds, providing insights into the structure of locally nonforesty graphs.

Q: What if we relax the condition of containing a cycle to containing a specific subgraph other than a cycle? How would the minimum size change?

Relaxing the condition to containing a specific subgraph H instead of a cycle would significantly impact the minimum size: H is a Forest: If H is a forest (e.g., a tree), the problem becomes trivial. A graph with no edges is locally foresty (all local subgraphs are empty graphs, which are forests). Thus, the minimum size would be 0. H Contains a Cycle: If H contains a cycle, the problem becomes more interesting. The minimum size would depend on the structure of H: Dense H: If H is dense (e.g., a clique), the minimum size would likely be larger compared to the cycle case. Ensuring that each local subgraph contains a dense H would require more edges. Sparse H: If H is sparse (e.g., a long cycle or a path), the minimum size might be smaller than the cycle case. It could be possible to construct sparser graphs where local subgraphs contain H. Forbidden Subgraph Characterization: The problem can be viewed through the lens of forbidden subgraphs. A graph is locally-H-free if no local subgraph contains H. Finding the maximum size of a locally-H-free graph is equivalent to finding the minimum size of a graph where every local subgraph contains H. In general, the minimum size would depend on: Structure of H: The density, connectivity, and other properties of H would influence the minimum size. Connectivity Requirement: The k-connectivity requirement would further constrain the possible constructions. This generalization opens up a rich area of research, exploring the interplay between local subgraph containment, forbidden subgraphs, and graph size.

Conceitos essenciais

This research paper disproves a conjecture about the minimum size of 3-connected locally nonforesty graphs and determines the exact minimum size.

Resumo

Bibliographic Information: Li, C., Tang, Y., & Zhan, X. (2024). The minimum size of a 3-connected locally nonforesty graph. arXiv preprint arXiv:2410.23702.
Research Objective: This paper investigates the minimum size (number of edges) of a 3-connected locally nonforesty graph, aiming to either prove or disprove a conjecture presented by Chernyshev, Rauch, and Rautenbach. A locally nonforesty graph is one where the neighborhood of every vertex induces a subgraph containing a cycle.
Methodology: The authors employ a combinatorial approach, utilizing proof by contradiction and case analysis. They first establish a lower bound for the size of such graphs. Then, they disprove the existing conjecture by demonstrating that the bound is not tight. Finally, they construct families of graphs achieving the exact minimum size for all orders greater than or equal to 8, thereby determining the true minimum size.
Key Findings: The paper disproves Conjecture 1 proposed by Chernyshev, Rauch, and Rautenbach, which stated that the minimum size of a 3-connected locally nonforesty graph of order n is at least 7(n-1)/3. The authors prove that the minimum size is significantly lower than this conjecture and provide an exact formula for the minimum size based on the graph's order.
Main Conclusions: The authors successfully determine the minimum size of a 3-connected locally nonforesty graph of order n. The result is presented as a piecewise function, f(n), dependent on the residue class of n modulo 8. The construction of specific families of graphs demonstrates the tightness of this bound.
Significance: This paper contributes significantly to the field of graph theory, specifically to the study of extremal graph theory and structural graph theory. By determining the precise minimum size of 3-connected locally nonforesty graphs, the authors provide a valuable tool for further research in this area.
Limitations and Future Research: The paper focuses specifically on 3-connected graphs. Future research could explore similar questions for graphs with different connectivity requirements or investigate the minimum size of locally nonforesty graphs with additional properties.

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Estatísticas

The minimum size of a locally nonforesty graph of order 5 is 9.
For a 3-connected locally nonforesty graph of order 100, the conjecture proposed a lower bound of 43 edges larger than the actual minimum size.

Citações

"A graph G is called locally nonforesty if every local subgraph of G contains a cycle."
"It turns out that Conjecture 1 does not hold."

Principais Insights Extraídos De

The minimum size of a $3$-connected locally nonforesty graph

by Chengli Li, ... às arxiv.org 11-01-2024

https://arxiv.org/pdf/2410.23702.pdf

The minimum size of a $3$-connected locally nonforesty graph

Perguntas Mais Profundas

How does the minimum size change if we consider k-connected locally nonforesty graphs for k > 3?

Finding the minimum size of a k-connected locally nonforesty graph for k > 3 becomes more intricate.  Here's why:

Increased Complexity: Higher connectivity (k > 3) imposes stronger restrictions on the graph's structure. The techniques used in the paper for 3-connected graphs, like analyzing the neighborhoods of minimum degree vertices and using the degree-sum formula, might not directly generalize to higher connectivity.

Dependence on Subgraph Structure: The requirement for each local subgraph to contain a cycle heavily influences the minimum size. As k increases, the local subgraphs need to be denser to ensure k-connectivity while still containing cycles. This interplay between connectivity and the cycle condition makes the problem significantly harder.

Potential for New Constructions:  The paper provides constructions for 3-connected locally nonforesty graphs. For k > 3, we might need entirely different construction techniques to achieve minimum size, potentially involving more complex base graphs and connection patterns.
In summary, determining the minimum size for k > 3 is an open problem that would likely require new methods and a deeper understanding of the interplay between connectivity, local subgraph structure, and forbidden subgraphs.

Could there be a relationship between the size of a locally nonforesty graph and its diameter or other graph invariants?

Yes, there could be relationships between the size of a locally nonforesty graph and other graph invariants like diameter:

Diameter and Size:  Locally nonforesty graphs with small diameters tend to have larger sizes. A small diameter enforces a certain density, as vertices need to be "close" to each other, leading to more edges.  Conversely, larger diameters might allow for sparser constructions while still maintaining the locally nonforesty property.

Other Invariants:

Girth: The girth (length of the shortest cycle) of a locally nonforesty graph is related to its size. Smaller girth often implies a larger size, as more edges are needed to create short cycles within each local subgraph.

Minimum Degree:  As seen in the paper, the minimum degree plays a crucial role in bounding the size. Higher minimum degrees generally lead to larger sizes in locally nonforesty graphs.

Independence Number:  The independence number (size of the largest independent set) could be inversely related to the size. A large independent set might make it harder to ensure the locally nonforesty property without adding many edges.
Exploring these relationships could involve:

Finding bounds:  Derive bounds on the size of locally nonforesty graphs based on their diameter, girth, or other invariants.
Characterizing extremal graphs:  Identify graphs that achieve equality in these bounds, providing insights into the structure of locally nonforesty graphs.

What if we relax the condition of containing a cycle to containing a specific subgraph other than a cycle? How would the minimum size change?

Relaxing the condition to containing a specific subgraph H instead of a cycle would significantly impact the minimum size:

H is a Forest: If H is a forest (e.g., a tree), the problem becomes trivial.  A graph with no edges is locally foresty (all local subgraphs are empty graphs, which are forests). Thus, the minimum size would be 0.

H Contains a Cycle: If H contains a cycle, the problem becomes more interesting. The minimum size would depend on the structure of H:

Dense H:  If H is dense (e.g., a clique), the minimum size would likely be larger compared to the cycle case.  Ensuring that each local subgraph contains a dense H would require more edges.

Sparse H: If H is sparse (e.g., a long cycle or a path), the minimum size might be smaller than the cycle case.  It could be possible to construct sparser graphs where local subgraphs contain H.

Forbidden Subgraph Characterization:  The problem can be viewed through the lens of forbidden subgraphs.  A graph is locally-H-free if no local subgraph contains H.  Finding the maximum size of a locally-H-free graph is equivalent to finding the minimum size of a graph where every local subgraph contains H.
In general, the minimum size would depend on:

Structure of H: The density, connectivity, and other properties of H would influence the minimum size.
Connectivity Requirement:  The k-connectivity requirement would further constrain the possible constructions.
This generalization opens up a rich area of research, exploring the interplay between local subgraph containment, forbidden subgraphs, and graph size.