Math: Graph Theory
Euler's Theorem
a graph must be connected and have EXACTLY zero or two odd vertices in order to be traceable. MEMORIZE WORD FOR WORD
Hamiltonian Graph
a graph that has a hamiltonian circuit
Eulerian graph
a graph that has an Euler circult or a graph with all even vertices.
Complete Graph
a graph where every pair of vertices are joined by an edge. K>n denotes the complete graph with n vertices.
Hamilton Circuit
a hamilton path that begins and ends at the same vertex.
Hamilton Path
a path that passes through all the vertices of a graph exactly once.
what does it mean to "eulerize" the graph?
add edges so all vertices even and we can find an Euler circuit
Leonhard Euler
altered the original question. rephrases question as "can i trace this graph?" Which means - can I begin at some vertex and draw the entire graph w/o lifting my pencil or going over any edge more than once?
edges
connect the vertices that have a bridge between them
traceable
draw without picking up pencil over every edge, drawing over no edge more than once. Euler says you don't have to end where you start.
Chromatic Number
either 2, 3, or 4, no bigger. The least number of colors required to do a color of a map/graph.
difference between euler path and euler circuit
euler path - regular traceable def. euler circuit - traceable, starts and ends in the same place.
every connected graph is eulerian - true or false?
false
connected graph
graph is connected if its possible to travel from any vertex to another vertex by moving along successive edges. not connected if its in two pieces. must be one piece.
Weighted Graph
graph whose edges have been assigned a number
in the complete graph Kn, there are (n-1)! ____ circuits.
hamilton
anytime we have a complete graph, we find a _____ circuit.
hamilton circuit
Path
in a graph, a series of consecutive edges in which no edge is repeated
the degree of every ______ is one less than the complete graph number after K.
less
factorial
n! K4 = (n-1) * (n-2) * (n-3) = 3*2*1
length of a path
number of edges in a path
A graph is not traceable if all vertices are ____.
odd
Euler Path
path that consists of all edges of the graph
vertices
plural.
graph theory is all about ______
relationshipss
vertex
singular. segment of a section of the city.
Francis Guthrie
student of DeMorgan. Asks DeMorgan: using at most 4 colors, is it always possible to color a map so that any two regions sharing a common border recieve different colors? DeMorgan asks Sir William Hamilton of Trinity college the same question.
degree of vertex
the number of edges that come out of the vertex. if the number is odd, vertex is odd. number is even, vertex is even.
weight of a path
the weight of a path is a weighted graph to the sum of the weight of the edges of the path.
In the Traveling Salesman problem, aka Cyndi Lauper touring problem, our goal is to minimize the total ____ of the ____ circuit.
total weight of the hamilton circuit
If a graph can be traced, then it has an Euler path. true or false?
true
Four Color Conjecture
using at most four colors, it is always possible to color a map so that any 2 regions sharing a common border recieve different colors. This was discovered after a 100 yrs by a computer program designed by Appel and Kaken in 1976 at University of Illinois. Some mathematicians considered it cheating. Since they have "proof" it now becomes the four color theorem. vertices = regions. edges connect vertices if the regions that the vertices represent share a border.
if all vertices are even, the connected graph can be traced at any ___. and you will start and end at the same place.
vertex
Difference between the original question and when euler rephrases it:
Euler doesn't say you have to end at the point where you started.
Brute Force Algorithm Steps
1 - list all possible hamilton circuits 2 - calculate weight of each circuit 3 - choose circuit w/ smallest total weight
Original Question
Can a person start in one section of the city, and cross each bridge exactly once and return to where they started?
