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burton_discrete_graph theory

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Transcript

  • 1. Graph Theory Chapter 3 By: Amber Dejha Sahar Isiah Kamal
  • 2. 3.1 Graphs, Puzzles, and Map Coloring
  • 3. Graphs
    • Graph- consist of finite set of point called vertices and lines called edges, that join pairs of vertices
    • Connected graph- if it is possible to travel from any vertex to any other vertex of the graph by moving along successive edges
    • Bridge- connected graph is an edge such that if it were removed the graph is no longer connected
  • 4.
    • Connected graph
          • Disconnected Graph
  • 5. How to trace a graph
    • Begin at some vertex and draw the entire graph without lifting your pencil and without going over any edge more than once
    • Graphs link!
  • 6. 4 Color Problem Theorem
    • Using at most 4 colors
    • 2 regions sharing a common border receive different colors
  • 7. 3.2 Hamilton & Circuit Paths
  • 8. Euler Theorem
    • A graph can be traced if…….
    • It is connected
    • It has either no odd vertices or two odd vertices
    • If it has 2 odd vertices, the tracing must begin at one of these and end at the other.
    • If all the vertices are even, then the tracing must begin and end at the same vertex .
  • 9. Euler
    • Path- A path in a graph is a series of consecutive edges in which no edge is repeated. The number in a path is called its length.
    • Euler Path- A path containing all the edges of a graph.(tracing)
  • 10. Examples CEFCDAFGABC Euler Circuit- An Euler path that begins and ends at the same time vertex
  • 11. Eulerian Graph- A graph with all even vertices. Eulerizing A Graph- Duplicate some edges in a graph to make all the vertices even . Euler Graph
  • 12. Hamilton Path
    • Hamilton Path- A path that passes through all the vertices of a graph exactly once is called a Hamilton Path.
    • Hamilton Circuit- A Hamilton Path that begins and ends at the same vertex .
  • 13. Graph
    • Complete Graph- A graph in which every pair of vertices is joined by an edge.
    • Weighted Graph- A graph that has numbers assigned to every edge.
  • 14.
    • Weights- Weights are the number of edges in a graph.
    • Weight of a Path- The Sum of the weights of the edges of the path.
    click here for live example of Hamilton graph!
  • 15. Pop Quiz
    • Determine whether each multigraph has an Euler path. Write yes or no.
    • 1) 2) 3)
    • Determine whether each multigraph has an Euler circuit. Write yes or no.
    • 1) 2) 3)
    YES NO YES YES NO YES