16planar_10 - ENGG1007 oundations of Computer Science...

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Unformatted text preview: ENGG1007 oundations of Computer Science Foundations of Computer Science Graphs Graphs Graph Coloring Professor Francis Chin and Dr SM Yiu November 25/26, 2010 Chapter 9.8 1 ENGG1007 FCS ap Coloring ap Coloring Map Coloring Map Coloring Given a map, try to color every region of the map such that adjacent regions have different colors. Can we color the map with the least number of colors? Four Color Problem All maps can be colored with 2 no more than four colors . ENGG1007 FCS roper Coloring roper Coloring Proper Coloring Proper Coloring Graph coloring (vertex)– color the vertices of the graph such that no two adjacent vertices are with the same color ap coloring olor the regions of a map such that no two Map coloring – color the regions of a map such that no two adjacent regions are of the same color Edge coloring – color the edge of a graph such that no two adjacent edges are of the same color Graph coloring should be the most general problem. hy? 3 Why? ENGG1007 FCS ap Coloring and Graph Coloring ap Coloring and Graph Coloring Map Coloring and Graph Coloring Map Coloring and Graph Coloring Map coloring Graph coloring Model this as a graph coloring problem (dual graph): Vertex – region Edge – connects regions that share borders Coloring – assignment of colors to vertices so that o two adjacent vertices are assigned the same color. 4 no two adjacent vertices are assigned the same color. Map coloring can be reduced to a “planar” graph coloring. ENGG1007 FCS dge Coloring and Graph Coloring dge Coloring and Graph Coloring Edge Coloring and Graph Coloring Edge Coloring and Graph Coloring Edge coloring can also be reduced to graph coloring, but ot the other way round not the other way round. Vertex corresponds to every edge of the original graph. Edge between two vertices when the corresponding edges g p g g are incident with the same vertex in the original graph. a d a c d ot possible the other way round by considering S b c e f b e f Not possible the other way round by considering S 3 S 3 does not have a corresponding graph for edge oloring, because it is not possible to have 3 edges 5 coloring, because it is not possible to have 3 edges adjacent to an edge but not to each other. ENGG1007 FCS ow Many Colors? ow Many Colors? How Many Colors? How Many Colors? Some graphs require fewer colors on lanar graphs may require more than 4 colors Non-planar graphs may require more than 4 colors Example: How many colors are required? C 6 K 6 K 3,4 Chromatic number Chromatic number of a graph is the least number of colors needed for coloring a graph....
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