16planar_10

# 16planar_10 - ENGG1007 Foundations of Computer Science...

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1 ENGG1007 Foundations of Computer Science Graphs Graphs Graph Coloring Professor Francis Chin and Dr SM Yiu November 25/26, 2010 Chapter 9.8

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2 ENGG1007 FCS 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 no more than four colors .
3 ENGG1007 FCS Proper Coloring Proper Coloring Graph coloring (vertex)– color the vertices of the graph such that no two adjacent vertices are with the same color 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. Why?

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4 ENGG1007 FCS Graph coloring Map Coloring and Graph Coloring Map Coloring and Graph Coloring Map 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 no two adjacent vertices are assigned the same color. Map coloring can be reduced to a “planar” graph coloring.
5 ENGG1007 FCS Edge Coloring and Graph Coloring Edge Coloring and Graph Coloring Edge coloring can also be reduced to graph coloring, but not the other way round. Vertex corresponds to every edge of the original graph. Edge between two vertices when the corresponding edges are incident with the same vertex in the original graph. Not possible the other way round by considering S 3 S 3 does not have a corresponding graph for edge coloring, because it is not possible to have 3 edges adjacent to an edge but not to each other. a b c d e f a b c d e f

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6 ENGG1007 FCS How Many Colors? How Many Colors? Some graphs require fewer colors Non-planar graphs may require more than 4 colors Example: How many colors are required?
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## This note was uploaded on 02/10/2011 for the course ENGG 1007 taught by Professor Unknown during the Spring '11 term at HKU.

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16planar_10 - ENGG1007 Foundations of Computer Science...

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