1. Euler and Hamilton's Paths and circuits

2. Euler Paths • An Euler path is when a trail on a graph visits each edge exactly once. • An Euler path must have an odd amount of degrees, and if the Euler is connected and has an even amount then it has at least one Euler circuit. • If you can start at a vertex and move to every single edge, it is an Euler path. • If something has more than 2 vertices of odd degree then it cant have an Euler path. • Sum of all degrees of all the vertices must be an even number

3. Euler Circuits • Euler circuit is much like the Eulerian path, but you start at one vertex, visit every vertex exactly once, then end on your starting vertex • They were first discussed by Leonhard Euler • A Euler circuit isn't a Euler path but A Euler path is a Euler circuit. • If every vertex is an even degree it is an Eulerian graph.

4. Hamilton Path • Hamilton’s path is a path in an undirected graph that visits every vertex once • Hamilton paths were named after William Rowan Hamilton who invented the Icosian game which is finding a Hamilton path in a dodecahedron. • A Hamilton’s path is also known as a Traceable graph.

5. Hamilton Circuit • Repeated traversals are not permitted • A Hamilton circuit is a circuit that visits every vertex once without touching and vertices more than once. • Hamilton circuit is also known as Hamiltonian graph. • The circuit returns to the starting point of the circuit

6. Questions • How do we use paths and circuit in the real world? • People who drive cars need to know how to get places and use the least amount of gas. • Why do we need to know about paths and circuits? • We need to know about paths and circuits in order to make things and in order to have electricity. • How are Hamilton’s path and circuit and Euler’s path and circuit alike and different? • They both need to go over a specific part of a graph. • They are different because one goes over edges while the other goes over vertices