M122GraphExSolns

# M122GraphExSolns - Exercise Set 1 1 b The shortest a g path has length 2 the longest has length 8 Of the 14 a g paths 1 has length 2 1 has length 3

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Exercise Set 1 . 1 b) The shortest a - g path has length 2; the longest has length 8. Of the 14 a - g paths: 1 has length 2, 1 has length 3, 1 has length 4, 2 have length 5, 4 have length 6, 3 have length 7, and 2 have length 8. (c) There are 7 a - g walks of length 4. Only 1 of these is a path. (d) There are 22 paths of length 2 in G , if the reverse of a path is regarded as the same path (i.e. a, b, c is the same as c, b, a ). Otherwise there are 44. (e) 3 : a, b, c, a ; 4 : a, c, d, b, a ; 5 : b, d, e, f, g, b ; 6 : b, d, e, f, h, g, b ; 7 : a, c, d, e, f, g, b, a ; 8 : a, c, d, e, f, h, g, b, a. 2 (a) ( n 2 ) (b) n (c) n - 1 (d) mn (e) 12 3. For any n 2 the graph P n has no cycles, but has the closed walk v 0 , v 1 , . . . , v n , v n - 1 , v n - 2 . . . , v 0 containing every vertex. 4. Consider a shortest closed trail of positive length containing x , say x 0 , x 1 , x 2 , . . . , x n , x 0 where x 0 = x . If this is a cycle, then there is a cycle containing x . Otherwise it is not a cycle, so it contains a repeated vertex. Thus there exists integers

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## This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.

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M122GraphExSolns - Exercise Set 1 1 b The shortest a g path has length 2 the longest has length 8 Of the 14 a g paths 1 has length 2 1 has length 3

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