I can Recognize an Euler path or Euler Circuit in a graph. 1. How do I recognize an Euler path in a graph? 2. How do I recognize an Euler circuit in a graph? Leonhard Euler By Samantha Lauderdale, DWE, VSD

Have you ever solved a maze? How did you find your way through it? Finding an Euler circuit or path in a graph is like solving a maze!

Linda is selling girl scout cookies. She wants to visit every street in her neighborhood, and end up back where she started. Which of these is a route that would be an EULER CIRCUIT? A B C D F G H a) A,B,C,D,E,F,G,H,C,B,A b)A,B,H,C,D,E,F,C,B,G,A c)A,B,C,D,E,F,C,H,G,F,H,B, G,A d)A,G,B,H,F,E,D,C,B,A E LET’S TRY ROUTE A! NO! It doesn’t work because we touched the same edge twice! LET’S TRY ROUTE B! NO! It doesn’t work because we missed three edges! LET’S TRY ROUTE C! X X X X X X X X X X X X X YES! It works! We touched each edge once, and finished where we started.

Luke lost his library book somewhere at school. He needs to visit each hallway once to look for it. What is a possible Euler path that he can take through the school? A B C D E F G H I a) G,H,F,E,D,C,B,I,H,A,B b) B,C,D,E,F,G,H,A,B,I,F,H,I,D c) H,F,E,D,C,B,A,H,G,F,I,B d)G,F,E,D,C,B,A,H,I,F,G,H LET’S TRY ROUTE A! NO! It doesn’t work! We missed three edges! LET’S TRY ROUTE B! YES! It works! We touched each edge exactly once!

Ellen’s mom needs to go to several stores in town. She wants to go down each road only once, and finish back at her house, where she started. Which of these possible routes would be an Euler Circuit? D DDD D D DD A B C D E F G a)F,G,H,B,A,E,D,C,B,A,E,F b)H,G,F,E,A,B,C,D,E,A c) F,E,D,E,A,B,H,G,F,E,B,C,D d) A,B,C,D,E,F,G,H,B,E,A H LET’S TRY ROUTE A! NO! It doesn’t work because we touched an edge more than once! LET’S TRY ROUTE B! NO! It doesn’t work because we touched an edge more than once! LET’S TRY ROUTE C! X X X NO! It doesn’t work because we touched an edge more than once! LET’S TRY ROUTE D! YES! It works because we touched each edge once, and finished where we started!

Tyler baked some cookies for his neighbors. He wants to go down each street in his neighborhood only once. What is a possible path that he can take to deliver his cookies? A BC D EF a) C,D,E,F,A,B,C b) E,F,A,D,C,B,A c) D,C,B,A,D,E,F,A d) A,B,C,D,E,F,A LET’S TRY ROUTE A! NO! It didn’t work because we didn’t touch one edge! LET’S TRY ROUTE B! NO! It didn’t work because we didn’t touch one edge! LET’S TRY ROUTE C! X X X X X X X YES! It worked because we touched each edge once!

How does solving a maze relate to recognizing an Euler Path or circuit in a graph? Write the answer on your slate!