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.
Read more about this topic: Extremal Graph Theory
Famous quotes containing the words minimum, degree and/or conditions:
“After decades of unappreciated drudgery, American women just dont do housework any morethat is, beyond the minimum that is required in order to clear a path from the bedroom to the front door so they can get off to work in the mourning.”
—Barbara Ehrenreich (20th century)
“One who shows signs of mental aberration is, inevitably, perhaps, but cruelly, shut off from familiar, thoughtless intercourse, partly excommunicated; his isolation is unwittingly proclaimed to him on every countenance by curiosity, indifference, aversion, or pity, and in so far as he is human enough to need free and equal communication and feel the lack of it, he suffers pain and loss of a kind and degree which others can only faintly imagine, and for the most part ignore.”
—Charles Horton Cooley (18641929)
“Each victim of suicide gives his act a personal stamp which expresses his temperament, the special conditions in which he is involved, and which, consequently, cannot be explained by the social and general causes of the phenomenon.”
—Emile Durkheim (18581917)