## What is a real life application of Euler paths and circuits?

Euler paths and Euler circuits are used in the real world by postmen and salesmen when they are planning the best routes to take. There can be multiple routes that they can take given a graph of the roads they need to pass by.

### Can you have a Euler path and circuit?

An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An Euler circuit starts and ends at the same vertex.

#### What is Euler path example?

One example of an Euler circuit for this graph is A, E, A, B, C, B, E, C, D, E, F, D, F, A. This is a circuit that travels over every edge once and only once and starts and ends in the same place.

**How do you identify Euler paths?**

Euler paths are an optimal path through a graph. They are named after him because it was Euler who first defined them. By counting the number of vertices of a graph, and their degree we can determine whether a graph has an Euler path or circuit.

**How is the Euler circuit important to everyday life?**

Euler is everywhere! They can also be used to by mail carriers who want to have a route where they don’t retrace any of their previous steps. Euler circuits and paths are also useful to painters, garbage collectors, airplane pilots and all world navigators, like you!

## How do you find Euler path and circuit?

If a graph has more than 2 vertices of odd degree then it has no Euler paths. 2. If a graph is connected and has 0 or exactly 2 vertices of odd degree, then it has at least one Euler path 3. If a graph is connected and has 0 vertices of odd degree, then it has at least one Euler circuit.

### Is Euler path NP complete?

– Euler circuit is in P, but Hamiltonian circuit is NP-complete. – Shortest path between two points is computable in O(1112), but longest path is NP- complete.

#### What is the difference between a Euler circuit and a Hamiltonian circuit?

Important: An Eulerian circuit traverses every edge in a graph exactly once, but may repeat vertices, while a Hamiltonian circuit visits each vertex in a graph exactly once but may repeat edges.

**What type of functions should be for Euler’s theorem?**

There is a theorem, usually credited to Euler, concerning homogenous functions that we might be making use of. A homogenous function of degree n of the variables x, y, z is a function in which all terms are of degree n.

**What is the difference between Euler path and Euler circuit?**

1 The graph must be connected. 2 When exactly two vertices have odd degree, it is a Euler Path. 3 Now when no vertices of an undirected graph have odd degree, then it is a Euler Circuit.

## Does the Königsberg graph have an Euler circuit?

A graph has an Euler circuit if and only if the degree of every vertex is even. A graph has an Euler path if and only if there are at most two vertices with odd degree. Since the bridges of Königsberg graph has all four vertices with odd degree, there is no Euler path through the graph.

### How many odd vertices can a graph with an Euler path have?

Recall that a graph with an Euler path can have at most two odd vertices, while a graph with an Euler circuit must have none. Since we have four odd vertices in our graph, this means that there is no solution to our problem, since the number of odd vertices exceeds the permissible for either graph.

#### Is there an Euler path that crosses every bridge exactly once?

There will be a route that crosses every bridge exactly once if and only if the graph below has an Euler path: This graph is small enough that we could actually check every possible walk that does not reuse edges, and in doing so convince ourselves that there is no Euler path (let alone an Euler circuit).