Core Concepts

This research paper presents a new Ore-type sufficient condition for the existence of a spanning jellyfish in a 2-connected graph, proving that any 2-connected graph G on n vertices with σ2(G) ≥ (2n-3)/3 contains a spanning jellyfish.

Abstract

**Bibliographic Information:**Kim, J., Kostochka, A., & Luo, R. (2024, October 14). Ore-type conditions for existence of a jellyfish in a graph. arXiv.org. https://arxiv.org/abs/2404.00811v2**Research Objective:**This paper aims to establish an exact Ore-type bound that guarantees the existence of a spanning jellyfish in a 2-connected graph.**Methodology:**The authors employ a proof by contradiction. They assume a counterexample graph that satisfies the given Ore-type condition but does not contain a spanning jellyfish. By analyzing the properties of this counterexample, particularly focusing on L-maximal cycles and the structure of components outside these cycles, they derive contradictions, ultimately proving the theorem.**Key Findings:**The paper proves that for n ≥ 13, every 2-connected n-vertex graph G with σ2(G) ≥ (2n-3)/3 contains a spanning jellyfish. This result implies a minimum degree condition: every 2-connected n-vertex graph G with δ(G) ≥ (n-1)/3 contains a spanning jellyfish. Additionally, the paper strengthens a previous result by Chen et al. for n ̸= 2 (mod 3), showing that if G is a connected, n-vertex graph with σ2(G) ≥ (2n-3)/3, then G contains a spanning broom.**Main Conclusions:**The paper provides an exact Ore-type degree condition for the existence of a spanning jellyfish in a 2-connected graph. This result contributes significantly to the field of graph theory, particularly in the study of spanning substructures and Hamiltonian properties of graphs.**Significance:**This research enhances the understanding of the relationship between the degree conditions and the presence of specific spanning subgraphs in graphs. It provides new insights into the structural properties of graphs and has potential applications in areas such as network design and algorithm optimization.**Limitations and Future Research:**The paper focuses specifically on spanning jellyfish and brooms. Exploring similar Ore-type conditions for the existence of other types of spanning subgraphs could be a potential direction for future research. Additionally, investigating the sharpness of the obtained bounds for all values of n could further strengthen the results.

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arxiv.org

Stats

n ≥ 13
σ2(G) ≥ (2n-3)/3
δ(G) ≥ (n-1)/3

Quotes

"The famous Dirac’s Theorem states that for each n ≥ 3 every n-vertex graph G with minimum degree δ(G) ≥ n/2 has a hamiltonian cycle."
"Gargano, Hell, Stacho and Vaccaro proved that every connected n-vertex graph G with δ(G) ≥ (n − 1)/3 contains a spanning spider, i.e., a spanning tree with at most one vertex of degree at least 3."
"Chen, Ferrara, Hu, Jacobson and Liu proved the stronger (and exact) result that for n ≥ 56 every connected n-vertex graph G with δ(G) ≥ (n − 2)/3 contains a spanning broom, i.e., a spanning spider obtained by joining the center of a star to an endpoint of a path."

Key Insights Distilled From

by Jaehoon Kim,... at **arxiv.org** 10-15-2024

Deeper Inquiries

It's certainly a natural question to explore! Here's a breakdown of the challenges and potential approaches:
Challenges:
Increased Complexity: As the connectivity (k) of a graph increases, the possible structures become more diverse and complex. This makes it harder to pinpoint precise degree conditions that guarantee specific spanning subgraphs like jellyfish.
Sharpness Examples: Finding sharp examples (graphs that barely fail the condition and don't have the desired subgraph) becomes trickier with higher connectivity. These examples are crucial for proving the tightness of any proposed degree bound.
Potential Approaches:
Inductive Arguments: One could try an inductive approach. Assuming the result holds for (k-1)-connected graphs, you'd aim to extend it to k-connected graphs. This might involve carefully analyzing how the addition of edges to increase connectivity affects the presence of jellyfish.
Structural Decomposition: Decompose a k-connected graph into "building blocks" of lower connectivity. If you can establish conditions for jellyfish within these blocks and how they combine, it might lead to a general result.
Relaxing the Structure: Instead of seeking a full spanning jellyfish, consider aiming for a "nearly spanning" jellyfish, where only a small number of vertices are outside the structure. This relaxation might make the degree conditions more manageable.
In Summary: Generalizing the Ore-type condition to k-connected graphs is a non-trivial problem. It would require a deep understanding of how higher connectivity influences the existence of spanning subgraphs and likely involve novel proof techniques.

Absolutely! Degree-based conditions are a natural starting point, but other graph parameters can provide valuable insights into spanning subgraphs. Here are some alternatives:
Connectivity-Related Parameters:
Toughness: Measures the resilience of a graph to vertex removal. A higher toughness might imply the existence of spanning jellyfish.
Binding Number: Quantifies the minimum ratio of neighbors within a set to the size of the set itself. A large binding number suggests good expansion properties, potentially favoring spanning structures.
Distance-Based Parameters:
Diameter: The longest shortest path in the graph. A small diameter could make it easier to construct a spanning jellyfish.
Average Distance: A low average distance might indicate a well-connected graph, increasing the likelihood of spanning subgraphs.
Eigenvalue-Based Parameters:
Spectral Gap: The difference between the two largest eigenvalues of the adjacency matrix. A large spectral gap often corresponds to good expansion properties, which can be beneficial for finding spanning structures.
Combinations and Hybrids: It's worth exploring combinations of these parameters. For instance, a graph with high toughness and small diameter might be particularly amenable to containing spanning jellyfish.
Beyond Parameters:
Forbidden Subgraph Characterizations: Identify specific subgraphs whose absence guarantees the existence of a spanning jellyfish. This approach focuses on structural properties rather than numerical parameters.
In Conclusion: Exploring alternative graph parameters and conditions can lead to a richer understanding of when spanning jellyfish exist. These investigations might reveal unexpected connections between different graph properties and provide new tools for studying spanning substructures.

Research on spanning substructures, like the spanning jellyfish in this paper, has significant implications for real-world network optimization problems. Here's how:
1. Routing Algorithms:
Efficient Broadcasting: Spanning trees, a simpler form of the spanning structures discussed, are fundamental for broadcasting information in networks. Finding spanning trees with specific properties (e.g., low diameter) directly translates to faster information dissemination.
Robust Routing: The existence of multiple disjoint spanning jellyfish or similar structures can provide redundant paths in a network. If one path fails, communication can continue uninterrupted through alternative routes, enhancing robustness.
2. Communication Network Design:
Network Connectivity: Understanding the relationship between degree conditions and spanning substructures helps in designing networks with desired connectivity levels. This ensures reliable communication even if some nodes or links fail.
Resource Optimization: By guaranteeing the existence of efficient spanning structures, network designers can optimize resource allocation (e.g., bandwidth, power) without compromising connectivity or performance.
3. Other Applications:
Parallel Computing: Spanning trees and related structures are crucial for efficient task distribution and communication in parallel computing environments.
Social Network Analysis: Identifying influential nodes or communities within a social network can be facilitated by understanding the properties of spanning subgraphs.
Bioinformatics: Graph theory plays a role in analyzing biological networks (e.g., protein-protein interaction networks). Spanning structures can help identify key pathways or functional modules.
Example:
Consider a communication network where we want to minimize the maximum number of hops (edges) any message needs to traverse. Finding a spanning tree with low diameter directly addresses this problem. If the network has additional properties that guarantee a spanning jellyfish, we might find even more efficient routing strategies due to the jellyfish's structure.
In Essence: The theoretical insights gained from studying spanning substructures provide practical tools for optimizing real-world networks. By understanding the conditions that guarantee their existence, we can design more efficient, robust, and cost-effective systems across various domains.

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