Extremal Graph Theory - Minimum Degree Conditions

Minimum Degree Conditions

The preceding theorems give conditions for a small object to appear within a (perhaps) very large graph. At the opposite extreme, one might search for conditions which force the existence of a structure which covers every vertex. But it is possible for a graph with

edges to have an isolated vertex - even though almost every possible edge is present in the graph - which means that even a graph with very high density may have no interesting structure covering every vertex. Simple edge counting conditions, which give no indication as to how the edges in the graph are distributed, thus often tend to give uninteresting results for very large structures. Instead, we introduce the concept of minimum degree. The minimum degree of a graph G is defined to be

Specifying a large minimum degree removes the objection that there may be a few 'pathological' vertices; if the minimum degree of a graph G is 1, for example, then there can be no isolated vertices (even though G may have very few edges).

A classic result is Dirac's theorem, which states that every graph G with n vertices and minimum degree at least n/2 contains a Hamilton cycle.