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Discrete Mathematics with Graph Theory (3rd Edition) 383

Discrete Mathematics with Graph Theory (3rd Edition) 383 -...

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Chapter 13 381 " a B E b~f c~e d We draw an edge between two vertices if the corresponding edges of 1{ cross. The original plans can't be met because 1{ does not have a 2-coloring. In fact, we must delete two vertices in order to obtain a subgraph with a 2-coloring. On the left below, we show 1{ " {a, b} and a 2-coloring. The adjacencies CF (edge a) and CG (edge b) are not met, but all others are. D (c) On the right above, we draw 91 again, but as a plane graph achieved by pulling the white edges, d and e, outside the circle. On the left below, we draw this graph as a plane graph with straight edges. Finally, on the right, we show a corresponding floor plan.
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Unformatted text preview: D A4ft7 E ~ ~ 22. We draw the relationship graph 91 with the vertices of a Hamiltonian cycle-H, K, DR, LR, F R, BI, Ba, B2 around a circle and labeling interior edges a, b, c, d, e, f, g, h as shown on the left. We then draw a new graph, 1{, whose vertices are labeled a, b, c, d, e, f, g, h and an edge indicates that the corresponding edges of 91 cross. H K B2 DR LR a b c d e Bl FR In order to 2-color 1{, it is necessary to remove only one of b, f or h. On the left below, we show the graph 1{ " {b} and a 2-coloring. On the right, we redraw 91 " {b} as a planar graph by pulling red edges a, f, g outside the circle....
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