Graphs and Euler Circuits

Chromatic #

# of colors used

Four Graph Color Thereom

Every planar graph is 4-colorable

Path

a movement from one vertex to another

Weight

each edge is associated with a value called weight

Closed Path

ends at the same vertex at which you started

Weighted Graph

graph in which each edge is associated with a value called weight. the value can represent any quantity we desire

Nonplanar Graph Thereom

graph is nonplanar if and only if it has the utilities graph or K5 as a subgraph, or it has a subgraph that can be contracted to the utilities graph or K5

Planar Graph

graph that can be drawn so that no edges intersect each other (except at vertices)

Subgraph Thereom

if a graph G has a subgraph that is not planar, then G is also not planar. In particular, G contains the utilities graph or K5, as a subgraph, G is not planar

Edges

line segments within the graph

Edge Picking Algorithm

mark edge of smallest weight in the graph mark edge of next smallest weight in the graph, as long as it does not complete a cirucit and does not add a third marked edge to a single vertex continue proccess until you can no longer mark any edges then mark the final edge that completes the hamiltonian circuit

2-Colorable Graph Thereom

2- colorable if and only if it has no circuits that consist of an odd number of vertices

Euler Path Thereom

a connected graph contains an euler path if and only if the graph has 2 vertices of odd degree with all other vertices of even degree. furthermore, every euler path must start at one of the vertices of odd degree and end at the other.

Eulerian

a connected graph is eulerian if an only if every vertex of the graph is of even degree

Graph

a set of points called vertices and line segments called edges that connect vertices

Complete Graph

all edges connect

Connected Graphs

any vertex can be reached from any other vertex by tracing along edges

Fluerys Algorithm

can be used to find euler circuits in graphs with large # of vertices trial and error

The Greedy Algorithm

choose a vertex to start and follow edge with smallest weight travel along edges with smallest weight that connects vertex not yet visited. continue through all vertices return to starting vertex. this has us choose the cheapest option at every chance we get

Dirac's Thereom

n is the # of vertices in the graph. if every vertex has degree of at least n/2, then the graph must be hamiltonian

Equivalent Graphs

the edges form the same connection of vertices in each graph

Hamiltonian Circuits

the path visits each vertex once and returns to the starting vertex withough visiting any vertex twice

Vertices

the points in the graph

Euler Circuits

uses every edge but never uses the same edge twice the degree of every vertex would have to be an even #