Discrete Mathematics with Graph Theory (3rd Edition) 395

# Discrete Mathematics with Graph Theory (3rd Edition) 395 -...

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Unformatted text preview: Chapter 14 393 11. (a) The new bipartition sets Vi and Vi are defined as follows: Vi = VI U {y}; Vi = V2 U {x}. (b) Choose any edge ab in the perfect matching of g, where a E VI, b E V2. For a matching of 9 U {x, y}, take edges ax, by and all edges in the perfect matching of 9 except for abo We easily see that this is a perfect matching for 9 U {x, y}. 12. (a) [BB] By Proposition 14.4.3, if Kn has a perfect matching, then n must be even. Conversely, assume that n = 2m is even and label the vertices UI, U2, .•• ,U2m. Since every pair of vertices is joined by an edge, a perfect matching will be {UIU2,U3U4, •.. ,U2m-IU2m}. (b) The result is still true, and the proof of (a) works as long as we label the vertices carefully. For example, choose UI and U3 to be the endpoints of e, so that e is not required in the matching. If n = 2, the result is false-in fact, K2 " {e} has no edges! 13. (a) Each edge in a perfect matching uses two vertices, and the vertices can only be used once. Hence, (a) Each edge in a perfect matching uses two vertices, and the vertices can only be used once....
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