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# A6 - (b Show that a graph with 98 vertices cannot have 49...

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MATH 239 ASSIGNMENT 6 Due Friday, 25th June at 11:45 a.m. 1. [12 marks] For n a positive integer, deﬁne G n to be the graph whose vertex set is the set of binary strings of length n which begin with a ‘1’ and have exactly three blocks, where vertices u and v are adjacent if they diﬀer in exactly one position (i.e. changing exactly one bit, either from 0 to 1 or from 1 to 0, turns u into v ). For example, 11001 and 10001 are adjacent, whereas 11011 and 10001 are not. (a) Draw the graphs G 3 , G 4 and G 5 . (b) Determine the number of vertices p in G n . (c) How many vertices of G n have degree 4? (d) For which n 3 is G n regular? (e) For which n 3 is G n bipartite? (f) For each n , which vertices are in the same component as the vertex 100 · · · 01? Is G n connected? 2. [6 marks] (a) Are the following two graphs isomorphic? Give an argument for your answer. (b) Are the following two graphs isomorphic? Give an argument for your answer. 3. [8 marks] (a) Give an example of a 3-regular graph that has a bridge.
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Unformatted text preview: (b) Show that a graph with 98 vertices cannot have 49 vertices of degree 4 and 49 of degree 3. (c) Show that if G is a graph with 100 vertices, in which u and v have degree 3 and the other 98 vertices all have degree 4, then there must be a path from u to v in G . (d) Show that if G is a 4-regular graph then G cannot have a bridge. 4. [6 marks] Answer ONE of the following. (a) Show that in a graph with p vertices, p ≥ 2, there must be two vertices which have the same degree. ( Hint: if all vertex degrees are diﬀerent, what must the smallest and largest degrees be? Show that this cannot happen. ) OR (b) Show that if G is a connected graph and its longest path length is k , then any two paths P and Q in G of length k must have at least one vertex in common. ( Hint: if P and Q have no vertex in common, ﬁnd a longer path, which is a contradiction. )...
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