Discrete Math: Hamilton Circuits Objective: To introduce Hamilton Circuits Last class we talked about Euler Paths and Euler Circuits and how to identify optimize the graphs. We left off talking about the Bridges of Madison County problem. Number of edges that need to be crossed? Total cost of the trip The photographer needs to start and end in the same place. Need a Euler Circuit with no odd vertices. Had 4 odd vertices. I added another link between L and D and another link between R and B. 11x25 = 275 to cross each bridge once Plus another 2x25 = 50 for the two additional links representing retrace) Total: $325 The photographer has the freedom to choose any starting and ending point he choses. Need a Euler path with exactly 2 odd vertices. Had 4 odd vertices, so I added a link between R and B, converting those 2 odd vertices to even vertices. 11x23 =275 1 x 25 = 25 Total: $300 The photographer must start his trip a point B and end the trip at point L. Need a Euler path where B and L remain odd and are the only odd vertices. I added links between C and D, between R and A, to make R and D even. These changes made A and C odd, so make those vertices even again, I added a third link connecting A and C. 11x 25 = 275 3x 23 = 75

Hamilton Paths and Circuits A Hamilton Path Is a path that visits each vertex of the graph once and only once. A Hamilton Circuit Is a path that visits each vertex of the graph once and only once and ends at the starting point. Look at the following graphs: How are Euler paths and Hamilton paths related? Euler Circuit Euler Path Hamilton Circuit Hamilton Path a Yes No b c d e f Conclusion: The existence of an Euler path or Circuit tells us nothing about the existence of a Hamilton path or circuit.

Dirac’s theorem: If a connected graph has N vertices (N>2) and all of them have degree bigger or equal to N/2, then the graph has a Hamilton Circuit. A complete graph is a graph with N vertices in which every pair of distinct vertices is joined by an edge. Examples of complete graphs Number of Edges in 𝐾 𝑁 : 𝐾 𝑁 has N(N-1)/2 edges Can you tell how many Hamilton Circuits on in each figure above? Number of Hamilton Circuits in 𝐾 𝑁 : 𝑁−1 ! Distinct Hamilton circuits in 𝐾 𝑁 Hamilton Path Puzzles