Hamilton Paths and Circuits

A collection of vertices and edges where Vertex-Edge Graph A collection of vertices and edges where each edge connects at least one of the vertices

Edges Line segments or curves connecting two vertices All edges in this graph are shown in red EDGE

Vertices The points where edges meet All the vertices in this graph are shown in red VERTEX

Hamilton Path A Hamilton Path is a path that visits each vertex exactly one time. A Hamilton Path starts at one vertex and ends at a different vertex. Hamilton Path

Hamilton Path Start End

Hamilton Circuit A Hamilton Circuit is a path that visits each vertex exactly one time. A Hamilton Circuit starts and ends at the same vertex. Hamilton Circuit 7

Hamilton Circuit Start End

How to Describe a Hamilton Circuit or Path End Start A B C D E F G H I A > H > I > G > F > C > E > D > B > A

A Common Trick Question on a Test Is this description of a Hamilton Path or Circuit correct? A > H > I > B > F > C > E > D > A

A Common Trick Question on a Test The Answer – it’s incorrect Vertex B is listed twice! A > H > I > B > F > C > E > D > A In a Hamilton PATH each vertex can only be visited once = NO repeating letters In a Hamilton CIRCUIT only the START and END letters may be repeated.

Practice Identifying Hamilton Paths and Circuits B E F D G A C Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker and describe it using vertex letters Does your answer depend on which vertex you start at? 12

Practice Identifying Hamilton Paths and Circuits B E F D G A C Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker and describe it using vertex letters Example: A > B > C > D > E > F > G Does your answer depend on which vertex you start at? YES: No Path or Circuit if start at D or F PATH Only

Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? B A K F H G J Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker and describe it using vertex letters Does your answer depend on which vertex you start at? Why? 14

Practice Identifying Hamilton Paths and Circuits B A K F H G J Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker or describe it using vertex letters one example: A > K > B > C > D > E > F > G > H > J Does your answer depend on which vertex you start at? If it does, explain how it changes, starting at other vertices. YES: Path at only A, B, J PATH Only (must start or end at A)

Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? B C E D F H J G K Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker or describe it using vertex letters Does your answer depend on which vertex you start at? 16

Practice Identifying Hamilton Paths and Circuits B C E D F H J G K Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker or describe it using vertex letters Example: A > B > D > G > H > E > J > K > F > C > A Does your answer depend on which vertex you start at? No Both a CIRCUIT and PATH