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Homework 2 copy

# Homework 2 copy - (6 Show that if a simple graph G is...

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HOMEWORK 2 8 PROBLEMS DUE: WEDNESDAY, FEBRUARY 2, 2011 (1) Let G be a simple graph where the vertices correspond to each of the squares of an 8 × 8 chess board and where two squares are adjacent if, and only if, a knight can go from one square to the other in one move. What is/are the possible degree(s) of a vertex in G ? How many vertices have each degree? How many edges does G have? (2) Let G be a graph with n vertices and exactly n - 1 edges. Prove that G has either a vertex of degree 1 or an isolated vertex. (3) Prove that if a graph G has exactly two vertices u and v of odd degree, then G has a u, v -path. (4) Let G be a simple graph. Show that either G or its complement G is connected. (5) Are any of the graphs N n , P n , C n , K n and K n,n complements of each other?
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Unformatted text preview: (6) Show that if a simple graph G is isomorphic to its complement G , then G has either 4 k or 4 k + 1 vertices for some natural number k . Find all simple graphs on four and ﬁve vertices that are isomorphic to their complements. (7) The complete bipartite graphs K 1 ,n , known as the star graphs , are trees. Prove that the star graphs are the only complete bipartite graphs which are trees. (8) A graph G is bipartite if there exists nonempty sets X and Y such that V ( G ) = X ∪ Y , X ∩ Y = ∅ and each edge in G has one endvertex in X and one endvertex in Y . Prove that any tree with at least two vertices is a bipartite graph....
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