Winter 2009 Assignment7 Soln

# Winter 2009 Assignment7 Soln - MATH 239 Winter 2009...

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MATH 239 Winter 2009 Assignment 7 Solutions [TOTAL: 15 marks] 1. (a) (b) (c) (d) For each of the four graphs (a) to (d) above, is the graph planar? In each case, either draw a planar embedding of the graph, or prove that one does not exist. Solution. [7 marks] The graph in (a) is just K 3 , 3 since if we put the top left, bottom middle, and top right vertices in one group, and the remaining three vertices in another group, each vertex in one group is adjacent to each vertex in the other group, and no two vertices in one group are adjacent. K 3 , 3 is non-planar. (Alternatively, if you don’t recognise that this is K 3 . 3 , you can observe that it has girth 4. So, assuming the graph is planar, and putting q k ( p - 2) / ( k - 2) where k is the girth (and p the number of vertices, q the number of edges) gives a contradiction.) The graph in (b) is planar as it can be redrawn by pulling every second of the “diagonal” edges outside, as in the ﬁrst of the following graphs:

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The graph in (c) has the second graph shown above as a subgraph. Note that this is an edge subdivision of the graph in (a), which is
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## This note was uploaded on 08/09/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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Winter 2009 Assignment7 Soln - MATH 239 Winter 2009...

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