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# burton_discrete_graph theory

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### burton_discrete_graph theory

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