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The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open ...

Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as (1,0), and the second by ...

A simple graph with n>=3 graph vertices in which each graph vertex has vertex degree >=n/2 has a Hamiltonian cycle.

Let a graph G have graph vertices with vertex degrees d_1<=...<=d_m. If for every i<n/2 we have either d_i>=i+1 or d_(n-i)>=n-i, then the graph is Hamiltonian.

An graph edge of a graph is separating if a path from a point A to a point B must pass over it. Separating graph edges can therefore be viewed as either bridges or dead ends.

A clique of a graph G is a complete subgraph of G, and the clique of largest possible size is referred to as a maximum clique (which has size known as the (upper) clique ...

Let a graph G have exactly 2n-3 graph edges, where n is the number of graph vertices in G. Then G is "generically" rigid in R^2 iff e^'<=2n^'-3 for every subgraph of G having ...

Any continuous function G:B^n->B^n has a fixed point, where B^n={x in R^n:x_1^2+...+x_n^2<=1} is the unit n-ball.

The average distance between two points chosen at random inside a unit cube (the n=3 case of hypercube line picking), sometimes known as the Robbins constant, is Delta(3) = ...

The edge set of a graph is simply a set of all edges of the graph. The cardinality of the edge set for a given graph g is known as the edge count of g. The edge set for a ...