Ch. 15: Graph Theory Some practical uses Degree of separation- Hollywood, acquaintance, collaboration Travel between cities Konigsberg bridge Shortest path Least cost Schedule exams, assign channels, rooms Number of colors on a map Highway inspecting, snow removal, street sweeping Mail delivery Niche overlap- ecology Influence graphs Round-robin tournaments Precedence graphs

Collaboration graphs Bacon ## people , , , Erdos ## people , , , , ,591 83,

See book and written handouts on Graph Coloring, mailroute, and Konigsberg bridge

Euler paths and circuits- definitions Euler circuit – a simple circuit containing every edge of G Note: circuits start and end at the same point Euler path – a simple path containing every edge of G Practical applications of Euler circuits:

Konigsberg bridge Konigsberg bridge problem

B C D A

Are there Euler paths or circuits for these graphs? A A A B A B C C C BD C D D E F B E AB CD

… A B CAB A B C C D C D D

Q—When is there an Euler circuit or path? A connected multigraph has an Euler circuit iff each of its vertices has _______. A connected multigraph has an Euler path but not an Euler circuit iff it has exactly _____.

Does this graph have an Euler circuit or Euler path? (look at degrees)

Hamilton circuits and paths Just touch every vertex once and only once We are not concerned with traveling along each edge Practical applications of Hamilton paths and circuits:

Do these graphs have Hamilton paths or circuits? A AB A B CA C C B D CD DE FB E AB CD

Hamilton paths and circuits A A B CAB BC D D ECD

Hamilton paths and circuits A BA B A B C C D CD DEF G E

Hamilton paths and circuits

Traveling salesman- p. 845

use Brute force or nearest neighbor approximation

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