MIT6_042JS10_lec16_sol

# MIT6_042JS10_lec16_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 10 Prof. Albert R. Meyer revised March 8, 2010, 683 minutes Solutions to In-Class Problems Week 6, Wed. Problem 1. For each of the following pairs of graphs, either deﬁne an isomorphism between them, or prove that there is none. (We write ab as shorthand for a b .) (a) G 1 with V 1 = { 1 , 2 , 3 , 4 , 5 , 6 } , E 1 = { 12 , 23 , 34 , 14 , 15 , 35 , 45 } G 2 with V 2 = { 1 , 2 , 3 , 4 , 5 , 6 } , E 2 = { 12 , 23 , 34 , 45 , 51 , 24 , 25 } Solution. Not isomorphic: G 2 has a node, 2, of degree 4, but the maximum degree in G 1 is 3. (b) G 3 with V 3 = { 1 , 2 , 3 , 4 , 5 , 6 } , E 3 = { 12 , 23 , 34 , 14 , 45 , 56 , 26 } G 4 with V 4 = { a,b,c,d,e,f } , E 4 = { ab,bc,cd,de,ae,ef,cf } Solution. Isomorphic (two isomorphisms) with the vertex correspondences: 1 f, 2 c, 3 d, 4 e, 5 a, 6 b or 1 f, 2 e, 3 d, 4 c, 5 b, 6 a (c) G 5 with V 5 = { a,b,c,d,e,f,g,h } , E 5 = { ab,bc,cd,ad,ef,fg,gh,he,dh,bf } G 6 with V 6 = { s,t,u,v,w,x,y,z } , E 6 = { st,tu,uv,sv,wx,xy,yz,wz,sw,vz } Solution. Not isomorphic: they have the same number of vertices, edges, and set of vertex de- grees. But the degree 2 vertices of G 1 are all adjacent to two degree 3 vertices, while the degree 2 vertices of G 2 are all adjacent to one degree 2 vertex and one degree 3 vertex. Problem 2. Deﬁnition ?? . A graph is connected iff there is a path between every pair of its vertices. False Claim. If every vertex in a graph has positive degree, then the graph is connected. (a) Prove that this Claim is indeed false by providing a counterexample. Creative Commons 2010, Prof. Albert R. Meyer .

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2 Solutions to In-Class Problems Week 6, Wed. Solution.
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## This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec16_sol - Massachusetts Institute of...

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