Open Problem Garden - r-regular graphs are not uniquely hamiltonian. - Comments

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?

r-regular graphs are not uniquely hamiltonian.

Robert Samal — Tue, 24 Jul 2007 07:32:08 +0200

Author(s): Sheehan

Subject: Graph Theory » Basic G.T. » Cycles

Conjecture If is a finite -regular graph, where , then is not uniquely hamiltonian.