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Euler and Hamilton Paths/Circuits A ____________ is a figure made up of points_________ connected by non-intersecting edges. Look at the graph below: A ____________is the ______________ of two edges. A _____________ is ______ if it is connected to an __________ _______________________ A ____________ is _______ if it is connected to an _________ _________________________ An ________ joins any two _________. It can be __________ or ______________ Euler Path: a graph is an Euler path if it can be________________ _______________________________________________________ _______________________________________________________ Vertices may be passed through ________________________ The starting and ending points _________________________ Euler Circuit: a circuit is similar to a Euler path, except _________________________________________________ _________________________________________________ Examples of Euler paths and circuits: Trace each graph to determine if it is an Euler Path or an Euler Circuit, or neither state why. Ex 1: Ex 2: Ex 3: What is the relationship between the nature of the vertices (odd or even) and the kind of graph (path or circuit)? Complete the table below using the following graphs. Graph 1 Graph 4 Graph # Graph 3 Graph 2 Graph 5 number of odd vertices Graph 6 number of even vertices Is it a… Euler path = P Euler circuit = C Neither = N 1 2 3 4 5 6 Conclusions: Based on the observations of your table above: 1) A graph with all vertices being even contains an Euler _________ 2) A graph with ______ odd vertices and _____________ contains an Euler ________. 3) A graph with more than 2 _______ vertices does not contain an Euler _______________ To name a path or circuit you list the vertices ________________________. Example 1a: Name an Euler circuit B a) One possible solutions is C D, E, F, A, D, C, A, B, D A b) Can you find another one? D ________________________ F E Example 1b: Given A, B, E, F, B, C, D, F, E, D is this Euler path or circuit or neither? _______________________ How can you tell? Explain your answer _____________________________________________________ ____________________________________________________________________________________ Ex 2: Find an Euler circuit if possible, if not list an Euler path. _____________________________________ C B F E A D Hamilton Paths and Circuits A __________________________ is a continuous path that passes through every _______________once and only once. A __________________________ is a Hamilton path that begins and ends at the same vertex. (the starting /end vertex will be the only vertex touched twice. How is a Hamilton Path different from a Euler path or Circuit?__________________________________ ____________________________________________________________________________________ Example 3: Find a Hamilton Path __________________________________________ (you name them like an Euler path or a Euler Circuit.) M S R K A O T H