# Further Maths - Eulerian Trails

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/eulerian-trails/ · Updated: 2021-11-04 · Tags: further-maths, year-2, decision, school

## See Also
## Flashcards
##### What is a Eulerian trail??
A trail that goes along every edge exactly once.

##### Why isn't a trail that visits every edge multiple times a Eulerian trail??
Because it visits an edge more than once.

##### What is a Eulerian trail that joins up to the beginning called??
A Eulerian cycle.

##### How many edges does entering a vertex and then exiting a vertex "use up" when constructing a Eulerian trail??
2

##### Why must all vertices be even for a Eulerian trail to be possible??
So that there is always a new path in and out of a vertex.

##### What is the technical condition for a graph to contain a Eulerian trail??
The order of all the vertices are even.

##### What does it mean for a graph to be "Eulerian"??
There is a Eulerian trail on the graph.

##### What does it mean for a graph to be "semi-Eulerian"??
It is possible to cover all edges without repeats starting at one vertex and ending at another.

##### What is the technical condition for a graph to be "semi-Eulerian"??
It must have all even vertex orders apart from two odd vertices.

##### What is the technical condition for a graph to be traversable??
It is Eulerian or semi-Eulerian.

##### What does it informally mean for a graph to be traversable??
You can draw the graph without taking your pencil off the paper and without repeating any edges.

##### What was the Konigsberg bridge problem??
Trying to cross each bridge exactly once and trying to end up back where you started.

##### What was the technique used to solve the Konigsberg bridge problem??
Representing it as a graph and showing it is not Eulerian.

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