Frequently Asked Questions

The problem is the Seymour Second Neighborhood Conjecture. It has been a major open question in mathematics and computer science, bridging these two disciplines. It involves oriented graphs and questions whether a node exists that has at least as many second neighbors as first neighbors. Solving this problem has implications for understanding graph theory, algorithm design, and many real-world applications like network analysis.

This problem has been studied for decades, 34 years to be exact. It has attracted attention from mathematicians and computer scientists alike. Its importance lies not only in its theoretical significance but also in its potential practical applications.

The problem has been tackled by numerous researchers. There have been significant partial results published and alternative conjectures proposed. However, the lack of a comprehensive approach incorporating both global and local perspectives likely limited progress until now.

This proof uses a global approach rather than a local one, made possible by establishing the well-ordering property. Then. instead of trying to guess where the node would double, we could employ an exhaustive search algorithm.

The well-ordering property ensures that any subset of elements has a minimum. For this problem, we had to first establish partitions, and an order with the distance relationship. With that and the minimum degree node as our anchor, we now have a least element.

Seymour originally asked if a node's degree would double in an oriented graph. The decreasing sequence property looks at what happens if this is not true - namely, it says that a nodes first neighborhood is strictly greater than its second neighborhood.

The proof is specific to this problem. However, the techniques—especially the use of the Graph Level Order (GLOVER) data structure has applications in other areas of mathematics, computer science, and graph theory.

My proof rigorously shows that no counterexamples can exist under the given conditions. On this website, I also include a section discussing possible refutations and why they do not hold.

The most challenging part was coming up with my own definitions. Graph theory is a field with over 200 years of history and a wealth of established literature. Writing formally while staying consistent with existing terminology was crucial, but introducing new ideas meant I often couldn't rely on references to explain what I was doing.

I had to define those terms myself, and that's a tricky process. On one hand, the definitions needed to be rigorous and clear enough to be understood by others. On the other hand, they had to align with the context of my research without causing confusion or conflicting with existing concepts. It was a balancing act between innovation and clarity, and sometimes I had to refine my definitions multiple times before they resonated with readers. Honestly. it's a walk I'm still trying to walk.

Feedback is crucial. I tested my definitions by explaining them to peers. If others could understand and use them to follow my arguments, I knew I was on the right track.

Absolutely. It was a game changer, especially for concepts like containers. At first, the idea seemed simple—essentially, an induced subgraph G[x]. But as I tried to describe their behavior in a network with specific properties, like interior and exterior arcs, and the next induced subgraph, the explanations became cumbersome.

I realized that using existing terminology wasn't capturing what I needed to express. So, I decided to coin my own term—“containers”—and commit to it. This shift made it significantly easier to talk about the structures I was analyzing. Once I had a clear term, it streamlined my thinking and communication, allowing me to focus on the deeper properties and relationships within the graph.

I think of it like building a card pyramid. Everyone wants to reach the top and create something grand. I wanted to practice patience, understanding that each individual card—the foundation—must be placed carefully and deliberately. Starting with small graphs, much like how Kaneko and Locke's paper, allowed me to build a solid foundation before scaling up. It's about ensuring that every step is well-grounded before attempting the complexity of larger graphs.

While it might seem tempting to dive into larger graphs right away, doing so can make it difficult to uncover the core principles that underlie the structure. By focusing on smaller graphs, I was able to refine my approach and test my ideas. This ensured that when I did scale up, I had a strong, stable framework to work with.

I am preparing the paper for submission to a mathematical journal. Details will be provided once it is accepted.

While the proof itself is rigorous and adheres to strict mathematical standards, the process that led to it was not purely mathematical. The key insight came from a computational perspective. An algorithm revealed the underlying structure of the problem. This algorithm provided the intuition to establish the core mathematical properties: the decreasing sequence property and the Graph Level Order (GLOVER) data structure.

Thus, the solution bridges mathematics and computer science. The algorithm highlights the computational nature of the problem, while the proof validates its correctness. I would refrain from labeling it "purely" anything. It's a synthesis of computer science and mathematical ideas.

The proof builds on foundational concepts in graph theory while addressing gaps in prior approaches. By utilizing divide and conquer in graph theory, I believe this connects a gap in that literature. Nodes in a graphs can no longer be seen as arbitrary points in space. Instead, with the GLOVER data structure, we can partition theese nodes. That's just one example.

The problem has implications for understanding algorithms, data structures, and computational complexity, particularly in the context of graph-based problems. In a large sense it shows that there are problems that we as computer scientists can be solving in the mathematical world. We can be creating new data structures and algorithms to tackle them.

While the proof is primarily theoretical, its insights into oriented graphs. The algorithm can be applied to various domains including social media, epidemiology, and security. Second, the GLOVER data structurecan can influence influence an even broader class of areas like network optimization, algorithm design, and complexity theory.

Yes! The website includes interactive visualizations and explanations designed to make the proof more accessible to those without a deep mathematical background.

While solving the SSNC, I noticed that most research papers addressing the conjecture were published in combinatorics or general mathematics sections. There was very little representation in computer science-focused venues. This surprised me because the conjecture sits at the heart of both disciplines. It is inherently mathematical but also deeply algorithmic, involving graph structures that computer scientists regularly work with.

Looking forward, I think part of the issue lies in how problems are categorized and communicated. Problems like the SSNC are often framed purely as mathematical challenges, which can limit interdisciplinary collaboration. Yet, these problems offer opportunities for unifying the approaches of math and computer science, leading to deeper insights and practical algorithms. My hope is that my work not only solves the SSNC but also highlights the importance of interdisciplinary thinking for tackling similar open problems in the future.

I've always believed that mathematics, when possible, should be taught with pictures. In this proof, I introduced my own notation and way of working with graphs in the paper. While it doesn’t change the proofs themselves, I wanted to do everything I could to make sure these definitions and concepts were as clear as possible. Visualization became a critical tool for achieving that clarity.

To start at the beginning, we'll start at the minimum degree node. Now I start my papers with some initial lemmas. Those were some of my first walks through the world of oriented graphs.

It wasn't long after that that I came up with the definition of decreasing sequence property. And that's when things started to get interesting. I'm drawing graphs, and I'm seeing patterns, but I need to test more cases. What about back arcs? Can I predict this?

I found that simply writing a computer program was easier than doing each example by hand. These first examples were on the JavaScript canvas. They also included an information table to let me know about the nodes.

Doing those programs helped my thinking evolve. Once I saw certain patterns, I knew what I was looking for, and looking to prove. The proofs still weren't easy, but I wasn't lost in the wilderness.

This work represents a significant personal achievement for me, and I want to ensure it’s not just shared but truly understood. A traditional paper has limitations — it’s static, bound by strict formatting rules, and offers no room for clarification or elaboration once published. What if a reviewer misinterprets an arrow in a definition, or a key visualization isn’t clear enough? I don’t want to risk my work being dismissed or misunderstood due to something I could have addressed with better tools.

A website allows me to take every step necessary to ensure my work is received as I intend. I can use animations, interactive diagrams, and explanatory notes to make every concept crystal clear. Visitors can interact with the ideas, explore visualizations at their own pace, and even ask questions if something isn’t clear. This isn’t just about presenting my work; it’s about starting a dialogue. I want to anticipate misunderstandings and prevent them before they happen.

By creating a space where my ideas can be explored dynamically and revisited as they evolve, I’m giving myself the best chance of ensuring this achievement is recognized and appreciated.

I welcome critiques as part of the scientific process. My website includes a section addressing common objections and potential counterexamples, explaining why they do not hold under the framework of my proof.

The proof has been carefully reviewed and tested against known results. Once published, it will undergo peer review by experts in the field. Additionally, the website provides interactive tools to help others understand and verify key concepts.

The journey spanned several years, starting with a desire to learn more about oriented graphs. The original breakthrough came with the decreasing sequence property. The final breakthrough came when I discovered the oriented complete graph inside a container. This led to multiple degree doubling nodes, a true statement of the Seymour Second Neighborhood Conjecture in abundance.

No, my initial goal was to explore oriented graphs and understand them better. The solution emerged naturally as I delved deeper into the problems.

With a PhD in Applied Mathematics and a strong background in computer science, I simply went towards what I enjoyed. I enjoy both fields, and I have no problem serving as a bridge between the two fields.

While the problem is technical, its solution demonstrates the power of human creativity and persistence. The methods used to solve it could inspire new ways of thinking in diverse fields. There are also practical applications of the Seymour Conjecture to things like social media and computer networking. The GLOVER data structure extends this beyond the proof and the problem itself to the concept of a total order in graph theory which has applications in many applications in the real world.

The website includes interactive tools and visualizations that explain key ideas in simple terms. These resources are designed to make the proof accessible to a wide audience.

Yes, the concepts I’ve developed have potential applications in graph theory and data science. The data structure used to represent these containers has been recognized for its novelty. I currently have a patent pending for it. This highlights not only the originality of the research but also its potential for practical use in applications that require efficient and dynamic graph representations.

One of the most frustrating aspects of research is the sheer difficulty of discovering open problems. It's easy to hear about famous conjectures like Goldbach’s Conjecture, the Collatz Problem, or Twin Primes. These are widely known and have a clear place in public discourse. But the reality is that there are hundreds of thousands of other problems. All these other problems are equally deserving of attention, but they aren't as well-known or discussed.

The challenge is not just finding these problems, but understanding their potential and how they fit into the broader research landscape. Many of these problems are deeply technical or niche. Without the right networks, it's easy to miss them. This creates a barrier for people who may be interested in contributing to new discoveries but aren’t aware of the problems that could use their skills and fresh perspectives.

There are platforms that are beginning to address this issue. For example, sites like Open Problem Garden and Wikipedia's Unsolved Problems in Mathematics. These platforms offer a space where researchers can find a collection of open problems that aren’t always part of the mainstream discussion but are valuable in their own right. If more people were aware of these resources and made an effort to explore them, we might see a greater interest in solving these problems and more opportunities for collaboration across disciplines.

I have provided the comments section of this page for that reason and encourage users to share their thoughts, questions, or concerns. If you find something unclear, have a question, or believe there might be an error, I would appreciate hearing from you. Engaging in a dialogue not only helps clarify misunderstandings but also strengthens the work through collective input.

To ensure your feedback is as actionable as possible:

Clearly reference the specific part or concept you’re addressing.
Explain what you find unclear or incorrect.
If possible, suggest a clarification or alternative interpretation.
This approach will help me respond effectively. It will also foster a collaborative environment where ideas can be refined together. Thank you for taking the time to engage with my work—your input is invaluable.

Yes, the website includes some computational examples and visualizations. These tools allow users to interact with the concepts and verify key properties.

I encourage others to explore the implications of this proof and test its ideas in related areas. Collaboration and further research will be key to fully understanding its impact.

Future work may explore how the methods used in this proof can be generalized to other problems in mathematics and computer science. I’m also interested in collaborating with others to find practical applications of these results

Check the website regularly for updates on the publication process, new resources, and potential collaborations.

Yes, the GLOVER data structure is covered by a provisional patent application filed with the U.S. Patent and Trademark Office. This ensures the innovation is protected while we continue to refine and develop its potential.