Let g be a graph containing a cycle c

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A. Let d ∈ N and V := {0, 1} d ; thus, V is the set of all 0-1 sequences of length d. The graph on V in which two such sequences form an edge if and only if they differ in exactly one position is called the d-dimensional cube. Determine the average degree, number of edges, diameter, girth and circumference of this graph. (Hint for the circumference: induction on d).

B. Let G be a graph containing a cycle C, and assume that G contains a path of length at least k between two vertices of C. Show that G contains a cycle of length at least √ k. Hint [1]: Consider how the path intersects C. Where can you see cycles, and can they all be short?

Reference no: EM13926662

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