A Brief Tour of Vertex-Edge Graphs
Focus Questions:
 What is a vertex-edge graph?
 What are some applications?
 What are some properties?
 What are some common and powerful types of problems?
1. Tracing puzzles (e.g., house puzzle)
Lead to recognition and definition of:
 vertex (vertices) and edge(s)
 vertex-edge graph (aka network, graph, finite graph, discrete graph)
 degree, circuit
2. Examples of some applications of vertex-edge graphs
 Document of images (available at www.infinitemath.com/mathforall/Home.html – see
vertex-edge graph resources in the Geometry section)
 www.visualcomplexity.com
3. Recommendations for teaching vertex-edge graphs in high school
 NCTM’s Principles and Standards for School Mathematics – geometry strand – “use
vertex-edge graphs to model and solve problems”
 Achieve Expectations – discrete mathematics strand – finite graphs
 Geometry’s Future report – The study of vertex-edge graphs is recommended in the
proceedings of a conference held almost twenty years ago on Geometry’s Future
(Malkevitch 1991). The future is now!
4. Task (adapted from NCTM’s Principles and Standards for School Mathematics) – Finding
an Optimum Paved-Road Network
Minimum spanning tree
(available at www.infinitemath.com/mathforall/Home.html)
5. Task – Optimum Zoo Tour Schedule
Vertex coloring
(available at www.infinitemath.com/mathforall/Home.html)
6. Some properties of graphs
Challenge: Can you draw a graph with one odd-degree vertex?
Lead to conjectures about the number of odd-degree vertices in any graph.
Theorems:
 The sum of the degrees of all the vertices in a graph is twice the number of edges.
 The sum of the degrees of all the vertices in a graph is even.
 Every graph has an even number of odd-degree vertices.
Brief Tour of Vertex-Edge Graphs
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7. Consider two classic route problems:
a. Plow all the streets in a neighborhood most efficiently; start and end at the same
place.
b. Collect money from all the ATMs in a neighborhood most efficiently; start and end at
the same place.
Problem a requires using each edge exactly once  Euler circuit
Problem b requires visiting each vertex exactly once  Hamilton circuit
8. Some problems, conjectures, theorems about Euler and Hamilton circuits
Explore some graphs, look for Euler circuits. Conjectures?
Theorems:
 Euler: A connected graph has an Euler circuit iff every vertex is even.
 Hamilton: A connected graph has a Hamilton circuit iff ____? (No existing theorem.)
9. Recommended vertex-edge graph topics (and sample applications) for high school
mathematics:
 Navigating Through Discrete Mathematics in Grades 6–12 (NCTM, 2008)
 “Vertex-Edge Graphs: An Essential Topic in High School Geometry.” Mathematics
Teacher (October, 2008)
10. A few resources:
 Navigating Through Discrete Mathematics in Grades 6–12 (NCTM, 2008)
 “Vertex-Edge Graphs: An Essential Topic in High School Geometry.” Mathematics
Teacher (October, 2008)
 Units on vertex-edge graphs in Core-Plus Mathematics, Glencoe, 2009. (Unit 4 in Course
1, Unit 6 in Course 2)
 www.infinitemath.com/mathforall/Home.html – See vertex-edge graph resources in the
Geometry section.
Summary Questions:
 What is a vertex-edge graph?
 What are some applications?
 What are some properties?
 What are some common and powerful types of problems?
Brief Tour of Vertex-Edge Graphs
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