Abstract:

The classical global criteria for the existence of Hamilton cycles and paths only apply to graphs with large edge density and small diameter. In a series of papers together with N. Khachatryan I have developed some local criteria for the existence of Hamilton cycles in a connected graph, which are analogues of the global criteria due to Dirac, Ore and others. The idea was to show that the global concept of hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. This local approach gives the possibility to find new classes of graphs with Hamilton cycles which, in particular, also contain infinite subclasses of graphs with small edge density and large diameter.

I shall give a review of this topic and present some new results.