This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, hamiltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area.

http://www.mathcs.emory.edu/~rg/updating.pdf

Updating the Hamiltonian problem—a survey

This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains some material on closely related topics such as traceable, pancyclic and Hamiltonian connected graphs.

Recent Advances on the Hamiltonian Problem: Survey III

A look at cycles containing specified elements of a graph

We study the role of connectivity of communication networks in private computations under information theoretical settings in the honest-but-curious model. We show that some functions can be 1-privately computed even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be 1-privately computed on non-2-connected networks. Furthermore, we present a technique for simulating 1-private protocols that work on arbitrary (complete) networks on k-connected networks. For this simulation, at most $(1 - k/(n - 1)) \cdot L$ additional random bits are needed, where L is the number of bits exchanged in the original protocol and n is the number of players. Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks 1-privately, namely $\lceil (n - 2)/(k - 1) \rceil - 1$ random bits for networks consisting of n players.

R. Glas

Papers

Cycles and paths through specified vertices in k -connected graphs