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7.1 Introduction to Graph Theory MAT 225 Discrete Math
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Page 1: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

7.1 Introduction to Graph Theory

MAT 225 Discrete Math

Page 2: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

The Seven Bridges of Köninsberg

Page 3: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Graph

vertices

vertex

edges

Page 4: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Definitions

A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex, and each edge in the sequence is between its endpoints.

A trail is a walk with no repeated edges. A circuit is a trail starting and ending at the same vertex.

A trail (or circuit) that uses every edge in a graph is called Eulerian.

Page 5: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Questions

Does the graph representing the seven bridges of Köninsberg have a Eulerian trail?

In general, how do we determine if a graph has a Eulerian circuit or Eulerian trail?

Page 6: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

When does a graph have an Eulerian circuit?

This graph does not have a Eulerian circuit.

This graph does have an Eulerian circuit.

Page 7: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

When does a graph have a Eulerian circuit?

This graph does have an Eulerian circuit.

How could I convinceyou that this graphhas an Eulerian circuit?

I can show it to you!

Page 8: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 9: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 10: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 11: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 12: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 13: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 14: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 15: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 16: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 17: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 18: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 19: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Finding the Eulerian Circuit

Page 20: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Why does a graph have a Eulerian circuit (or not)?

Why does this graph not have a Eulerian circuit?

Why does this graph have a Eulerian circuit?

Page 21: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Degree

The degree of a vertex is the number of edges that meet at that vertex

Page 22: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Degree

The degree of a vertex is the number of edges that meet at that vertex

Page 23: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Degree

The degree of a vertex is the number of edges that meet at that vertex

For example, in thisgraph, the degree ofC is 4 because thereare four edges (to B,to F, to G, and to H)that meet there

Page 24: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

You might be able to tell right away that this graph can’t possibly have an Eulerian circuit

Why not?

Page 25: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

If a graph has a vertex with degree 1, the graph cannot have an Eulerian circuit If we start at E, we will

never be able to return to E without retracing

If we don’t start at E, when we go there we cannot leave without retracing.

Page 26: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

The problem isn’t just degree 1

This graph also doesn’t have an Eulerian circuit

The problem is that some of the degrees are odd numbers

Page 27: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

Let’s focus on vertex D, which has degree 5

Suppose we start elsewhere in the graph

Page 28: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

Since we want to cover all edges, we’ll have to visit D eventually

Page 29: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

We have several unused edges, so we need to follow one of them and leave D

Page 30: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

In fact, every time we visit a vertex, we will “use up” two of the edges that meet at that vertex

Page 31: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

We have unused edges, so we need to visit D again at some point…

Page 32: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

…and then leave again…

Page 33: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

…and then come back again. But now we’re stuck, since we can’t leave D

without retracing, but D wasn’t our starting point.

Page 34: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

What if we had started at D?

Page 35: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

First, we need to leave D…

Page 36: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

… then sometime later, we have to come back to D…

Page 37: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

… and then leave again …

Page 38: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

… and then come back again …

Page 39: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

What does degree have to do with Eulerian circuits?

… and then leave again.

But D was our starting point, and we have run out of edges to use to come back to D!

Page 40: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

No Eulerian Circuits With Odd Degrees

If a graph has any vertex with an odd degree, then the graph does not have an Eulerian circuit

The reverse is true as well

Page 41: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Theorem 2 (in section 7.1)

A graph G has a Eulerian circuit if and only if every vertex has an even degree.

Page 42: MAT 225 Discrete Math. vertices vertex edges A walk is in a graph is an alternating sequence of vertices and edges which begins and ends with a vertex,

Theorem 5 (in section 7.1)

A graph G has a Eulerian trail if and only if G has exactly two vertices with odd degree. Moreover, the trail must begin and end at these two vertices.


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