Hamilton Paths & Hamilton Circuits SOL: DM.2 Classwork Worksheet Homework (day 66)

“Thinking Mathematically” by Blitzer Pg: 842 We have studied paths and circuits that covered every edge of a graph. Using every edge of a graph is helpful for garbage collection, curb sweeping , or snow removal. What about a UPS driver trying to find the best way to deliver packages around town? In this situation a driver is only concerned about finding a path that passes through each vertex of the graph only once. Thus, it is not important to use every road or edge between the delivery locations.

Hamilton Path A path that passes all the vertices of a graph exactly once (must include all vertices).

Hamilton Circuit Starts at a vertex and visits each vertex once and only once, returning to where it started.

https://www.youtube.com/watch?v=AamHZhAmR7o 1 2 4 3

The number of Hamilton circuits in a complete graph If you have a graph that is complete and has a Hamilton circuit it has multiple sequences that will produce the same Hamilton circuit. In order to avoid a duplication you can start and end at vertex A. How many unique Circuits can you get from a complete Hamilton graph with three vertices? 2 A B C D

The number of Hamilton circuits in a complete graph The number of Hamilton circuits in a complete graph with n vertices. (n-1)! Determine the number of Hamilton circuits in a complete graph with: Four vertices b) Five Vertices c) Eight Vertices 6 24 5040

Ex: Find the Hamilton paths and/or circuits Ex: Determine whether the graph has a Hamilton Circuit. If so, find one.

Ex: For reach graph, some paths are specified. Highlight which paths are Hamilton Circuits. If not, justify. a) A→B→C→D→E→C→A→𝐸→𝐹→𝐴 b) A→C→D→E→F →𝐴 c) F→A→C→E→F d) C→D→E→F→A→𝐵 Ex: In the graph, a string of vertices is specified. Determine whether the string is a circuit, whether it is an Euler circuit and whether it is a Hamilton circuit. Justify. a) 𝐴→𝐵→𝐶→𝐷→𝐸→F→𝐺→𝐴 b) 𝐵→𝐼→𝐺→𝐹→𝐸→D→𝐻→𝐹→𝐼→𝐷→𝐶→𝐵→𝐺→𝐴→𝐵 c) A→𝐵→𝐶→𝐷→𝐸→𝐹→𝐺→H→𝐼→𝐴