Hi everyone,

I very recently started working on the Book Embedding Conjecture, which can be found on the Open Problem Garden website (which is now mysteriously down again after having gone back up briefly...you can still view it on Yahoo's cache). I got a great start on it today, and should be finished with the problem by tomorrow...maybe the day after if I'm slower or more busy than I expect to be.

-Philip White (philipjwhite@yahoo.com)

On Wednesday, June 8, 2022 at 7:41:55 PM UTC-4, B.H. wrote:
> Hi everyone,
>
> I very recently started working on the Book Embedding Conjecture, which can be found on the Open Problem Garden website (which is now mysteriously down again after having gone back up briefly...you can still view it on Yahoo's cache). I got a great start on it today, and should be finished with the problem by tomorrow...maybe the day after if I'm slower or more busy than I expect to be.
>
> -Philip White (philip...@yahoo.com)

I just found a proof that the Book Thickness of Subdivisions Conjecture, as stated on the Open Problem Garden, is false.

Also, the webpage asserts that the conjecture as stated is true for complete graphs, but that is not correct...from the paper:

"The main result of this paper, whose proof is based on Ramsey theory, states that every graph obtained from Kn and Kn;n by subdividing each edge at most once has large book-thickness when n is large."

I addressed the conjecture as stated on the OPG website, where it says, "Let G' be the graph obtained by subdividing each edge of G exactly once."

I will check my proof over more carefully and write everything I need to write out formally, and verify that the proof is correct or print a retraction within a day or two...it will probably be later tonight.

Here is a "challenge question" of my own that I know the answer to that is interesting to state:

Given a graph G and a given (not necessarily minimal/optimal) valid linear ordering of G, characterize the computational complexity in some good way for finding the minimal value of k for a valid k-page book embedding based on the given linear ordering?

As I said I can answer this question...I haven't checked the answer yet and won't publish the solution, but I'm posing the question for others who might be interested. I will confirm/retract that I have the solution to this problem that I've stated after I check my work on that too.

I think I am doing an excellent job at showcasing my highly excellent math theorem proving skills.

-Philip White (philipjwhite@yahoo.com)

On Thursday, June 9, 2022 at 1:26:41 PM UTC-4, B.H. wrote:
> On Wednesday, June 8, 2022 at 7:41:55 PM UTC-4, B.H. wrote:
> > Hi everyone,
> >
> > I very recently started working on the Book Embedding Conjecture, which can be found on the Open Problem Garden website (which is now mysteriously down again after having gone back up briefly...you can still view it on Yahoo's cache). I got a great start on it today, and should be finished with the problem by tomorrow...maybe the day after if I'm slower or more busy than I expect to be.
> >
> > -Philip White (philip...@yahoo.com)
> I just found a proof that the Book Thickness of Subdivisions Conjecture, as stated on the Open Problem Garden, is false.
>
> Also, the webpage asserts that the conjecture as stated is true for complete graphs, but that is not correct...from the paper:
>
> "The main result of this paper, whose proof is based on Ramsey theory, states that every graph obtained from Kn and Kn;n by subdividing each edge at most once has large book-thickness when n is large."
>
> I addressed the conjecture as stated on the OPG website, where it says, "Let G' be the graph obtained by subdividing each edge of G exactly once."
>
> I will check my proof over more carefully and write everything I need to write out formally, and verify that the proof is correct or print a retraction within a day or two...it will probably be later tonight.
>
> Here is a "challenge question" of my own that I know the answer to that is interesting to state:
>
> Given a graph G and a given (not necessarily minimal/optimal) valid linear ordering of G, characterize the computational complexity in some good way for finding the minimal value of k for a valid k-page book embedding based on the given linear ordering?
>
> As I said I can answer this question...I haven't checked the answer yet and won't publish the solution, but I'm posing the question for others who might be interested. I will confirm/retract that I have the solution to this problem that I've stated after I check my work on that too.
>
> I think I am doing an excellent job at showcasing my highly excellent math theorem proving skills.
>
> -Philip White (philip...@yahoo.com)

I didn't do fully formal write-ups where I wrote out every step as I would if I were publishing, patenting, selling, or otherwise submitting my work, but I just finished confirming that both of my theorems (the OPG one and the answer to my own challenge problem that I presented) are proved correctly! I won't be publishing this work, but the second theorem in particular is a nice one.

I'll probably pick another problem to solve tomorrow.

-Philip White (philipjwhite@yahoo.com)

On Friday, 10 June 2022 at 04:04:06 UTC+8, B.H. wrote:
> On Thursday, June 9, 2022 at 1:26:41 PM UTC-4, B.H. wrote:
> > On Wednesday, June 8, 2022 at 7:41:55 PM UTC-4, B.H. wrote:
> > > Hi everyone,
> > >
> > > I very recently started working on the Book Embedding Conjecture, which can be found on the Open Problem Garden website (which is now mysteriously down again after having gone back up briefly...you can still view it on Yahoo's cache). I got a great start on it today, and should be finished with the problem by tomorrow...maybe the day after if I'm slower or more busy than I expect to be.
> > >
> > > -Philip White (philip...@yahoo.com)
> > I just found a proof that the Book Thickness of Subdivisions Conjecture, as stated on the Open Problem Garden, is false.
> >
> > Also, the webpage asserts that the conjecture as stated is true for complete graphs, but that is not correct...from the paper:
> >
> > "The main result of this paper, whose proof is based on Ramsey theory, states that every graph obtained from Kn and Kn;n by subdividing each edge at most once has large book-thickness when n is large."
> >
> > I addressed the conjecture as stated on the OPG website, where it says, "Let G' be the graph obtained by subdividing each edge of G exactly once."
> >
> > I will check my proof over more carefully and write everything I need to write out formally, and verify that the proof is correct or print a retraction within a day or two...it will probably be later tonight.
> >
> > Here is a "challenge question" of my own that I know the answer to that is interesting to state:
> >
> > Given a graph G and a given (not necessarily minimal/optimal) valid linear ordering of G, characterize the computational complexity in some good way for finding the minimal value of k for a valid k-page book embedding based on the given linear ordering?
> >
> > As I said I can answer this question...I haven't checked the answer yet and won't publish the solution, but I'm posing the question for others who might be interested. I will confirm/retract that I have the solution to this problem that I've stated after I check my work on that too.
> >
> > I think I am doing an excellent job at showcasing my highly excellent math theorem proving skills.
> >
> > -Philip White (philip...@yahoo.com)
> I didn't do fully formal write-ups where I wrote out every step as I would if I were publishing, patenting, selling, or otherwise submitting my work, but I just finished confirming that both of my theorems (the OPG one and the answer to my own challenge problem that I presented) are proved correctly! I won't be publishing this work, but the second theorem in particular is a nice one.
>
> I'll probably pick another problem to solve tomorrow.
>
> -Philip White (philip...@yahoo.com)

I think you are too brilliant. Don't publish your works (selling is worse),
it is too dangerous, it can trigger WW3. Many people would die.