I wanted to bring up my paper on the Seymour Second Neighborhood Conjecture. The #coordinatePlane and proof use a well ordering which used BFS and a minimum counterexample. #maths #math #graphtheory #mathisbeautiful #algorithm alphaxiv.org/abs/2501.00614
I’ll be giving a talk on my solution to SSNC at a Combinatorics Conference in March entitled
A Minimum Counterexample Proof of the SSNC via the Graph Level Order
What happens when they still do
Yesterday, five-year-old Liam and his dad Adrian were released from Dilley detention center. I picked them up last night and escorted them back to Minnesota this morning.
Liam is now home. With his hat and his backpack.
Rest easy, Demond Wilson
🥹🙏🏾💙🕊
Look for the helpers #MrRogers
And email. Teachers just send packets of busy work. Don't they realize that getting past level 4 on Mario Brothers is busy work. Learning to climb on top of walls and secret worlds. That's the perfect snow day activity.
1 Peter 5:8
Be alert and of sober mind. Your enemy the devil prowls around like a roaring lion looking for someone to devour.
1 Peter 5:8
Be alert and of sober mind. Your enemy the devil prowls around like a roaring lion looking for someone to devour.
How I approached the Seymour Second Neighborhood Conjecture. Standard visualizations lead to high crossing numbers, leading to difficulty in recognizing patterns. By utilizing a BFS layout, and a minimal criminal, patterns emerged. #math #graphtheory www.alphaxiv.org/abs/2501.00614
Am image of an oriented graph for the Seymour Second Neighborhood Conjecture.
This is one of my first illustrations for the Seymour Second Neighborhood Conjecture #SSNC. It's an oriented graph. I noticed patterns, number of nodes in each neighborhood. #MathIsBeautiful
1->7
2->6
3->5
Second, number of cycles.
1->1
2->2
3->2 (can't have 3)
alphaxiv.org/abs/2501.00614
This is an oriented graph for the Seymour Second Neighborhood Conjecture #SSNC. Nodes are in "rooted neighborhoods". We see patterns, number of nodes in each neighborhood. #MathIsBeautiful
1->7
2->6
3->5
Second, number of cycles.
1->1
2->2
3->2 (can't have 3)
alphaxiv.org/abs/2501.00614
This #ProofByContradiction is like building a house of cards. What we see is that: either the house will collapse, leading to a contradiction and producing that degree-doubling node, or we finish constructing the house, and every node fails to be a degree-doubling node. #math
What is this proof by contradiction?
We're assuming that the #SSNC is false. That means that every node in the graph has second neighborhood smaller than it's first neighborhood. Then we investigate the overall graph and see if it can still stand.
So what is this proof by contradiction?
It is a #BottomUpApproach. Starting with the #definitions. I realized that first and second neighbors are not good enough because you lose tracking info. Second neighbors do not know which first neighbor an edge came from. #research
I told the story about how excited I was with my initial results. The problem is, noone else was doing the problem #MyWay, which made collaboration difficult. This #KnowledgeGraph shows how the #SSNC problem has been #researched over the decades.
I'm excited to share my #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to #Research this #OpenProblem in #GraphTheory.
This second expert saw the same problem with missing cases, but constructed a more abstract example.
That was 5 years ago. And that's where more #Research and the need for a #dataStructure came in. It combats these missing cases, but also brought order to the overall problem.
I tightened up my definitions and my proofs, so I thought. I understood why the missing cases were always resolved immediately in concrete examples.
I was so confident, I sent the paper for review to another expert. And ...He saw the same problem with these missing cases.
In the coming weeks I contacted him, showed him a very preliminary draft of a paper I was writing, and a JavaScript program I had written for this research.
The main point was I thought I had solved the problem.
The expert was concerned I had missed some cases.
An expert was giving a talk on combinatorial game theory - a fascinating talk. After the talk, I rush up to him to tell him I've been working on a 30 year old problem of his. I was sure I embarrassed myself, but he gave me his email. #ResearchJourney #AcademicLife #Mentorship
I'm excited to share my #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to #Research this #OpenProblem in #GraphTheory.
Parent nodes have a burden: they do not want their degrees to double.
Children can assist by load balancing: forming cycles in the following neighborhoods.
The children also have this burden.
These dual burdens and load balancing shrink neighborhoods and yield a contradiction
Parent nodes have a burden: they do not want their degrees to double. Children can assist by load balancing: forming cycles in the following neighborhoods. The children also have this burden. These dual burdens and load balancing shrink neighborhoods and yield a contradiction
I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.
I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.
I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.
This image shows my early coding to try to solve the conjecture. What we see are The nodes of the same color in cycles, I called them interior neighbors. The nodes of different colors are I called exterior neighbors. Wanted to investigate this more. Not pictured here are back arcs.
I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.
My Biblical thought of the day, Jesus went through Samaria, a town that Jews would walk mikes to avoid. So after the Resurrection, did this continue? Did Christians like Paul continue looking for converts in Sameria?
