Hamilton Cycle Theorem?!

Question

I've come across this link, where the author states: "Hamilton Cycle Theorem fails for infinite graphs unless ..."

Please help me on this, what does he mean by "Hamilton Cycle Theorem"? I studied a whole chapter on hamilton cycles, have seen some necessary and sufficient conditions due to Ore, Dirac, Chvatal, Bondy, etc. but none of them titled or referred as "Hamilton Cycle Theorem"! What theorem does the author of the link examine?!

FYI, I've studied the popular book for undergraduates on Graph Theory by Bondy and Murty.

Answer 1

The slide set you link to does not say "Hamilton Cycle Theorem", but "Hamilton cycle theorems", uncapitalized and in plural. This difference is meaningful.

In other words, it is not speaking about one particular theorem, but about a vaguely defined class of theorems that have to do with Hamiltonian cycles -- presumably, in this context, theorems that give sufficient criteria for a graph to have a Hamiltonian cycle.

One concrete example of the theorems the author is speaking about appears on slide 9:

Theorem (Tutte ’56) Every finite 4-connected planar graph has a Hamilton cycle.

And a different one appears on slide 38:

Theorem (Fleischner ’74) The square of a finite 2-connected graph has a Hamilton cycle.