Open Problem Garden - Comments

New partial results (re: Chords of longest cycles)

Robert Samal — Sat, 14 Oct 2023 12:17:00 +0200

New partial results for this appear in

Carsten Thomassen: Chords in longest cycles, JCTB 129 (2018) 148-157

Possible solution (re: Do any three longest paths in a connected graph have a vertex in common? )

Robert Samal — Sat, 14 Oct 2023 12:13:51 +0200

Possible solution to this appears in https://arxiv.org/abs/2006.16245

(I am not sure whether it is being in a referee process or whether somebody may have found a gap in the presented proof.)

Context (re: 3-Edge-Coloring Conjecture)

Anonymous — Thu, 23 Jun 2022 16:27:00 +0200

Is this conjecture missing some greater context? It seems obviously false on its own

question (re: 3-Edge-Coloring Conjecture)

Anonymous — Wed, 22 Jun 2022 17:35:55 +0200

wouldn't removing any edge from a cubic graph make the graph not cubic?

The construction in that (re: r-regular graphs are not uniquely hamiltonian.)

Anonymous — Mon, 20 Jun 2022 14:23:36 +0200

The construction in that paper has parallel edges, so it is not a counter example. As far as I am aware, Sheehan's conjecture is still open.

The conjecture is only for (re: r-regular graphs are not uniquely hamiltonian.)

Anonymous — Tue, 03 May 2022 23:00:27 +0200

The conjecture is only for simple graphs. The paper you mention gives counterexamples that have multiple edges.

Is this question still open? (re: r-regular graphs are not uniquely hamiltonian.)

Anonymous — Thu, 03 Feb 2022 12:06:15 +0100

I think, the autor answers the question in this article: Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs (https://doi.org/10.1002/jgt.3190180503). (And the conjecture is fals for all even values.) Am I wrong?

This conjecture is false (re: Partition of Complete Geometric Graph into Plane Trees)

Anonymous — Thu, 06 Jan 2022 12:27:03 +0100

This conjecture has recently been disproved, see arXiv:2108.05159 and arXiv:2112.08456.

Poincare conj is true in some dimensions > 7. (re: Smooth 4-dimensional Poincare conjecture)

Anonymous — Thu, 26 Aug 2021 00:24:03 +0200

For n equals 12, 56 and 61, Sn also has a unique smooth structure.

A counterexample? (re: 3-Edge-Coloring Conjecture)

Anonymous — Tue, 03 Aug 2021 06:31:13 +0200

What would be the cubic graph homeomorphic to K4-e? I think I can show there does not exist a cubic graph homeomorphic to K4-e. If so, this would seem to contradict the conjecture's claim.

This problem is solved (re: Book Thickness of Subdivisions)

Anonymous — Sat, 10 Jul 2021 14:52:34 +0200

This problem is solved in:

V. Dujmović, D. Eppstein, R. Hickingbotham, P. Morin, D. R. Wood. Stack-number is not bounded by queue-number. Combinatorica, accepted in 2021 https://arxiv.org/abs/2011.04195

Linear transformation of primitive pythagorean n-tuple (re: Primitive pythagorean n-tuple tree)

Anonymous — Wed, 16 Jun 2021 20:36:19 +0200

Is it true that for primitive pythagorean n-tuple (a_1, a_2...... a_n), if there are (n-2) even terms and 1 odd term in the summation side of the equation and a_n is odd, a linear transformation can be made? And can we find any such primitive pythagorean n-tuple where a_n is even?

Problem statement (re: Jacobian Conjecture)

Anonymous — Thu, 04 Mar 2021 17:44:19 +0100

iff the determinant of the Jacobian matrix is a nonzero constant.

Already solved (re: Inscribed Square Problem)

Anonymous — Thu, 25 Feb 2021 17:01:43 +0100

This was proven in https://arxiv.org/pdf/2005.09193.pdf

Claimed to be solved (re: Complete bipartite subgraphs of perfect graphs)

Anonymous — Tue, 19 Jan 2021 14:32:20 +0100

In this recent preprint on Paul Seymour's webpage: http://web.math.princeton.edu/~pds/papers/pure5/paper.pdf (not on ArXiv nor published yet)