# burton_discrete_graphtheory

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## burton_discrete_graphtheoryPresentation 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
• 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.
• 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