Reference no: EM13926895
1. Show that the components of a graph partition its vertex set. (In other words, show that every vertex belongs to exactly one component.)
Hint: Assume the contrary, and derive a contradiction.
2 .Show that every 2-connected graph contains a cycle.
Hint: Find two vertices that are linked by two independent paths.
3. Determine k(G) and λ(G) for G = P m, Cn, Kn, Km,n and the d-dimensional cube (Exercise 2); d, m, n ≥ 3.
Hint: For each type of graph, the solution requires separate proofs of (coinciding) upper and a lower bounds. For the cube, use induction on n.
4. Is there a function f : N → N such that, for all k ∈ N, every graph of minimum degree, at least, f (k) is k-connected?
Hint: Try to find counterexamples for k = 1.
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Components of a graph partition its vertex set
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