Which of these sequences are graphic justify your

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Unformatted text preview: quence of decreasing integers is called graphic if it corresponds to the degrees of the vertices a graph. Which of these sequences are graphic? Justify your answer. (a) 3, 2, 2, 2, 1, 1 SOLUTION. Not a graphic sequence since the number of vertices of odd degree in a graph must be even. (b) 3, 2, 1, 1, ￿ SOLUTION. A graphic sequence. Let V (G) = {A, B, c, d, e} and 1 ￿ E (G) = {A, c}, {A, d}, {A, B }, {B, e} . (c) 7, 5, 3, 2, 1, 1, 1 SOLUTION. Cannot be graphic. In a graph with 7 vertices, the maximal degree of a vertex is 6. 3. Let Gn be the graph with vertex set {1, . . . , n} and such that u and v are adjacent if and only if u + v ≡ 3 mod 4. (a) Draw a drawing of G9 . SOLUTION. Here is one example: (b) Show that Gn is bipartite. SOLUTION. Let’s see what are all the possibilities, mod 4, to get 3 as a sum: 0 + 0 ≡ 0, 0 + 1 ≡ 1, 0 + 2 ≡ 2, 0 + 3 ≡ 3 , 1 + 1 ≡ 2, 1 + 2 ≡ 3 , 1 + 3 ≡ 0, 2 + 2 ≡ 0, 2 + 3 ≡ 1, 3 + 3 ≡ 2. So u + v ≡ 3 if and only if • u ≡ 0 and v ...
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This document was uploaded on 03/07/2014.

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