Hamiltonicity criteria for sparse graphs Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course).
There are three main classes of criteria for Hamiltonicity that I am aware of:


*

*Dirac-type conditions ($\delta \geq \frac{n}{2}$, i.e. high minimum degree).

*Spectral conditions.

*Erdos-Chvatal-type conditions ($\kappa \geq \alpha$, i.e. connectivity greater than independence number).
However, none of these approaches is able to settle even the Hamiltonicity of $C_{n}$, the $n$-cycle! Apparently, the reason is that these approaches work best for dense graphs.
Is there an alternative criterion that works well for sparse graphs?
 A: Partial answer.
According to Eppstein

It is known that it is NP-complete to test whether a Hamiltonian cycle exists in a 3-regular graph, even if it is planar (Garey, Johnson, and Tarjan, SIAM J. Comput. 1976) or bipartite (Akiyama, Nishizeki, and Saito, J. Inform. Proc. 1980) or to test whether a Hamiltonian cycle exists in a 4-regular graph, even when it is the graph formed by an arrangement of Jordan curves (Iwamoto and Toussaint, IPL 1994).

So in general you can't expect complete answer to your question.
Another reason for Hamiltonicity are forbidden subgraphs, which
force Hamiltonicity (possibly with a finite number of exceptions).
Not sure if this works for sparse graphs, perhaps search the web.
Found this: Forbidden subgraphs, hamiltonicity and closure
in claw-free graphs 
There are theorems of the form:

Every 2-connected $XY$-free graph is hamiltonian.

Where $X,Y$ are explicit small graphs.
For certain type of graphs polynomial time
algorithms exist for HC, so if the they work for sparse
graphs you can find a cycle, e.g. this paper
