Euler Tours

Neeldhara Misra

DSA1 Week 4

1

1

2

1

2

3

1

2

3

4

1

2

3

4

5

1

2

3

4

5

6

1

2

3

4

5

6

7

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Work through the exercises in the Mathigon lesson

about the Bridges of Königsberg

A walk is a sequence of edges
\(e_1, \ldots, e_{n-1}\)

 

such that there exists a sequence of vertices
\(v_1, \ldots, v_n\)

for which \(e_i = (v_i, v_{i+1})\)
for all \(1 \leqslant i \leqslant n-1\).

An (closed/open) Euler Tour of a graph \(G\)

is a (closed/open) walk

that contains every edge exactly once

(i.e, no repeats and no omissions).

If a graph \(G\) has:

 

(a) more than two vertices of odd degree, or

(b) exactly one vertex of odd degree, then

 

it does not have an Euler Tour of any kind

(closed or open).

What we discovered

If a graph \(G\) has:

 

(a) exactly two vertices of odd degree, or

(b) no vertex of odd degree, then

 

does it have an Euler Tour of some kind

(closed or open)?

Food for thought