graphs, networks, and finite geometries

vertices and edges

a graph is a collection of

euler tour

each edge is traced exactly once and the terminal node is the same as the initial node

euler path

each edge is traced exactly once

all vertices are even

the graph gives a tour if

two vertices are odd

the graph gives a path if

4 color theorem

every map can be colored using most 4 colors only

1st axiom of three-point geometry

there exist exactly three distinct points in the geometry

2nd axiom of three-point geometry

each two distinct points are on exactly one line

3rd axiom of three-point geometry

not all points of the geometry are on the same line

4th axiom of three-point geometry

each two distinct lines are on at least one point

1st axiom of four-point geometry

there exist exactly four distinct points in the geometry

2nd axiom of four-point geometry

each two distinct points are on exactly one line

3rd axiom of four-point geometry

each line is on exactly two points

Fano’s geometry

There exists at least one line

Every line has exactly three points on it.

Not all points of the geometry are on the same line.

Two distinct points are on exactly one line.

Each two lines have at least one point on both of them.

Fano’s geometry theorems

Each two distinct lines have exactly one point in common.

There exist exactly seven points and seven lines in Fano’s Geometry.

Each point lies on exactly three lines.

The lines through any point contains all the points of the geometry.

For any pair of points, there exist exactly two lines not containing either point.

young’s geometry

There exists at least one line.

Every line has exactly three points on it.

Not all points of the geometry are on the same line.

Two distinct points are on exactly one line.

If a point does not lie on a given line, then there is exactly one line on that point parallel to the given line