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.

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

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

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.

Read Job. So was Satan.

Who bears the brunt of these cuts? Well, here's a word cloud of the most frequent terms from the titles and abstracts of terminated grants, with word size proportional to frequency:

The problem is that there are too many people like Donald Trump voting.