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Dec 18, 2013 at 22:33 comment added Mohammad Al-Turkistany @MichalR.Przybylek I edited the question to restrict the scope and make it answerable
Dec 18, 2013 at 21:59 comment added Michal R. Przybylek @MohammadAl-Turkistany, I don't think you've asked a research-level question and I doubt it's possible to provide a research-level answer. Here is another great observation --- take the very interesting class of graphs that have at least one vertex of degree 1 and merge it with the very interesting class of graphs consisting of a single cycle.
Dec 18, 2013 at 21:47 comment added Mohammad Al-Turkistany I'm looking for non-silly classes :). I'm interested in interesting graph classes (similar to the ones in my post) with infinitely many Hamiltonian graphs and infinitely many non-Hamiltonian graphs. The same applies to the Hamiltonian path property.
Dec 18, 2013 at 21:09 history answered David E Speyer CC BY-SA 3.0