T6 - k 1 Problem 3 Let G n be the graph where the vertices...

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Problem 1: The complement of a graph G , denoted by G , is the graph with V ( G ) = V ( G ) and, for every u,v V ( G ) with u 6 = v , the edge { u,v } ∈ E ( G ) if and only if { u,v } 6∈ E ( G ). Prove that, if a graph is not connected, then it is not isomorphic to its complement. Problem 2: Let G be a graph with minimum degree at least k . Prove that G contains a path of length at least k and a cycle of length at least
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Unformatted text preview: k + 1. Problem 3: Let G n be the graph where the vertices consist of all binary strings of length n , and two strings are adjacent if they diﬀer in exactly two positions. Show that G n is not connected (for n ≥ 1), and not bipartite (for n ≥ 3). How many vertices and edges does G n have? 1...
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