Disc-a12

# Disc-a12 - a and b The&ve paths from a to b are a;b...

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1. Let K n be the complete graph on n vertices ( each two vertices are adja- cent ), and K m;n the complete bipartite graph : V = X [ Y where X \ Y = ; , j X j = m , j Y j = n , and E = ff x;y g : x 2 X and y 2 Y g . (a) How many vertices and edges does K 7 have ? The graph K 7 has 7 vertices and 7 2 ± = 21 edges. (b) What is ( K 10 ) ? The only way to color K 10 without having two adjacent vertices the same color is to use a di/erent color on each vertex, so ( K 10 ) = 10 . (c) How many vertices and edges does K 3 ; 4 have ? The graph K 3 ; 4 has 3 + 4 = 7 vertices and 3 4 = 12 edges. (d) What is ( K 3 ; 5 ) ? There are two adjacent vertices in K 3 ; 5 so one color is not enough. On the other hand, if you color the vertices in X red and the vertices in Y green, then no two adjacent vertices will be the same color, so ( K 3 ; 5 ) = 2 . (e) Label the vertices of K 4 . Pick two vertices and list all the paths from one of them to the other . Suppose we label them a;b;c;d and pick the vertices

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Unformatted text preview: a and b . The &ve paths from a to b are a;b a;c;b a;d;b a;c;d;b a;d;c;b (f) Label the vertices of K 3 ; 3 . Pick two vertices and list all the paths from one of them to the other . Suppose we let X = f a;b;c g and Y = f d;e;f g . The nine paths from a to b are a;d;b a;e;b a;f;b a;d;c;e;b a;d;c;f;b a;e;c;d;b a;e;c;f;b a;f;c;d;b a;f;c;e;b The nine paths from a to d are a;d a;e;b;d a;e;c;d a;f;b;d a;f;c;d a;e;b;f;c;d a;e;c;f;b;d a;f;b;e;c;d a;f;c;e;b;d 2. Construct a graph with 5 vertices and 6 edges that is not connected . The graph K 4 has 4 vertices and 6 edges. If we add an isolated vertex, we get a graph with 5 vertices and 6 edges that is not connected. e e e e e & & & @ @ @ ± ± ± H H H...
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## This note was uploaded on 07/13/2011 for the course MAD 2104 taught by Professor Staff during the Spring '08 term at FAU.

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Disc-a12 - a and b The&ve paths from a to b are a;b...

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