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Graph. Planar Graph. A graph is called planar if it can be drawn in the plane without any edges crossing A graph may be planar even if it is usually drawn with crossings because it may be possible to draw it in a different way without crossings. Is K 4 planar?
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Planar Graph • A graph is called planar if it can be drawn in the plane without any edges crossing • A graph may be planar even if it is usually drawn with crossings because it may be possible to draw it in a different way without crossings.
Is K4 planar? • K4 is planar because it can be drawn without crossings. • Is Q3 planar? • Q3 is planar, because it can be drawn without any edges crossing.
Graph Coloring • A coloring of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent vertices are assigned the same color. • A graph can be colored by assigning a different color to each of its vertices. What is the least number of colors necessary? • The chromatic number of a graph is the least number of colors needed for a coloring of this graph. denoted by X (G). (Here X is chi.)
What are the chromatic numbers of the graphs G and H • The chromatic number of G is at least three. • H has a chromatic number equal to 4.
What is the chromatic number of the complete bipartite graph Km,n , where m and n are positive integers? Ans. X (Km ,n ) = 2 • What is the chromatic no. of the graph Cn, where n >=3? Ans. X (Cn ) = 2 if n is an even positive integer with n>= 4 and X (Cn ) = 3 if n is an odd positive integer with n>=3 .
