**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, ï¿¿ SOLUTION. A graphic sequence. Let V (G) = {A, B, c, d, e} and
1
ï¿¿
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 â‰¡ 3 mod 4.
(a) Draw a drawing of G9 .
SOLUTION. Here is one example: (b) Show that Gn is bipartite.
SOLUTION. Letâ€™s see what are all the possibilities, mod 4, to get 3 as a sum:
0 + 0 â‰¡ 0, 0 + 1 â‰¡ 1, 0 + 2 â‰¡ 2, 0 + 3 â‰¡ 3 ,
1 + 1 â‰¡ 2, 1 + 2 â‰¡ 3 , 1 + 3 â‰¡ 0,
2 + 2 â‰¡ 0, 2 + 3 â‰¡ 1,
3 + 3 â‰¡ 2. So u + v â‰¡ 3 if and only if â€¢ u â‰¡ 0 and v â...

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