sol4 - AMS 301 Spring, 2011 Homework Set # 4 Solution Notes...

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AMS 301 Spring, 2011 Homework Set # 4 — Solution Notes #1, 2.3: (j). The graph is bipartite, only 2 colours are needed. (n). χ = 4. There is a wheel formed by nodes a,b,d,j,i,c, as in Example 2 so we need at least 4 colours. Since the graph is planar, 4 colours suFce, for example: a-2, b-4, c-3, d-1, e-3, f-1, g-2, h-4, i-1, j-2. #2, 2.3: (b). 3 colours are needed since nodes have degree 3. 3 colours are enough: ( a, d ), ( b, e ) and ( c, f )) colour 1, ( a, e ), ( b, f ) and ( c, d ) colour 2, and the remaining 3 edges colour 3. (d) By Vizing’s Theorem, we know we ned either 4 or 5 colours, since the max node degree is 4. In fact 4 colours are enough: Colour 1 are edges (e,a) (b,c) and (d,g). Colour 2 are edges (a,b) (e,g) (c,f). Colour 3 are edges (a,d) (b,e) (f,g). Colour 4 are edges (a,g) (c,d) (e,f). Note: the 4 colour Theorem is not helpful here, as it is about node colouring, not edge colouring. #10, 2.3:
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This note was uploaded on 05/28/2011 for the course AMS 301 taught by Professor Estie during the Spring '11 term at University of Florida.

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