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    burton_discrete_graph theory burton_discrete_graph theory Presentation Transcript

    • Graph Theory Chapter 3 By: Amber Dejha Sahar Isiah Kamal
    • 3.1 Graphs, Puzzles, and Map Coloring
    • 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
      • Connected graph
            • Disconnected Graph
    • 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!
    • 4 Color Problem Theorem
      • Using at most 4 colors
      • 2 regions sharing a common border receive different colors
    • 3.2 Hamilton & Circuit Paths
    • 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 .
    • 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)
    • Examples CEFCDAFGABC Euler Circuit- An Euler path that begins and ends at the same time vertex
    • 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
    • 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 .
    • 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.
      • 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!
    • 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