Eulerian Graphs (Edexcel IAL Maths)

Degree (Valency) of a Vertex

Spec point — 2.1

What is meant by the degree (valency) of a vertex?

• The degree or valency (or order) of a vertex (node) can be defined by how many edges are incident(connected) to it

  • A loop at a vertex increases the valency of a vertex by 2 as both ends of the edge are connected to it

• A vertex can be described as being odd or even

  • It has odd degree if there are an odd number of edge connections
  • It has even degree if there are an even number of edge connections

What is Euler's handshaking lemma?

• Euler's handshaking lemma states that for any undirected graph:

  • The sum of the degrees of the vertices is equal to twice the number of edges
  • The number of odd vertices must therefore be even (or zero)

Eulerian & Semi-Eulerian Graphs

What are Eulerian cycles and trails?

• An Eulerian cycle starts and ends at the same vertex and traverses every edge in a graph exactly once

  • Unlike a true cycle it may visit a vertex more than once
  • An Eulerian cycle is also known as an Eulerian circuit

• An Eulerian trail traverses every edge exactly once but starts and ends at different vertices

  • Again, vertices may be visited more than once

What are Eulerian and semi-Eulerian graphs?

• An Eulerian graph is a graph that contains an Eulerian cycle

  • Every vertex in an Eulerian graph has an even valency

• A semi-Eulerian graph is a graph that contains an Eulerian trail

  • Exactly one pair of vertices in the graph will have odd valencies

    • These odd vertices will be the start and finish points of any Eulerian trail

• Eulerian graphs can be used to solve many practical problems where the edges should not be traversed more than once

  • A common problem is the Chinese Postman problem

Exam Hint

• You can quickly tell if a graph is Eulerian or semi-Eulerian

  • Can you draw the graph without taking your pen off the paper and without going over any edge more than once?
  • If yes, then you have an Eulerian or semi-Eulerian graph!

Worked Example Let G be the graph shown below.

a) Show that G is a semi-Eulerian graph.

Answer:

Look at the degree of each vertex.

A: 2

B: 4

C: 3

D: 3

• E: 4 *

G is a semi-Eulerian graph because it has exactly one pair of odd vertices, C and D

b) Write down an Eulerian trail for G.

Answer:

An Eulerian trail must start and end at C/D

There are several possible Eulerian trails, one solution is

Final answer: DEABECDBC