Discrete Mathematics
Theorems on Euler Circuits and Euler Paths
Name: ________________________
Date: ______________________
Origin of Graph Theory
- Bridges of Konigsberg (1736)
 Mathematician Leonhard Euler visited the town of Konigsberg and was asked to solve a puzzle: “could a
person walk around the town in such a way that each of the seven bridges is crossed once, and only once?”
How can we draw a graph based on the Seven Bridges of Konigsberg?
Based on the graph that we drew, is it possible to cross each bridge once and only once?
Path vs. Circuit
How else can we apply the definitions of paths and circuits to a graph representing airline flights available
throughout the U.S.?
1. If Cleveland is a vertex, list all the other vertices.
Los
Angeles
2. If Chicago to Atlanta is an edge, list all the other edges
3. List all the paths from Los Angeles to Cleveland.
Chicago
Cleveland
St.
Louis
Houston
4. Give an example of a circuit that starts and ends in Cleveland with at most 4 stops.
Atlanta
Euler Circuit vs Euler Path
Last class we also learned the definitions of an Euler Circuit and an Euler Path.
Euler Circuit:
Euler Path:
These can now be extended into theorems.
Euler’s Theorem #1 – Circuit Theorem
a) If a graph has ____________________________,
b) If a graph is ___________________________ and every vertex is _______________, then it has at least one Euler Circuit.
Examples: Do the graphs below contain Euler circuits?
1)
2)
3)
4)
Euler’s Theorem #2 – Path Theorem
a) If a graph is connected and has exactly ____________________________________________, then it has at least one
__________________________. Any path must start at one of the odd vertices and end at the other one.
Examples: Do the graphs have an Euler Path?
1)
2)
3)
4)
Euler’s Theorem #3 – Vertex-Edge Theorem
a) The sum of the degrees of all the vertices of a graph equals _________________________ the number of edges
b) The graph always has an even number of ______________ vertices
Examples: How many edges are there in each graph?
7) deg(A) = 4, deg(B) = 2, deg(C) = 3, deg(D) = 3
8) deg(A) = 5, deg(B) = 4, deg(C) = 6, deg(D) = 1, deg(E) = 2
9) deg(A) = 10, deg(B) =6, deg(C) = 9, deg(D) = 7