Comportamiento Reológico de la Suspensión

Section 4.1. Connectivity and Edge-Connectivity

1. Determine the connectivity and edge-connectivity of each complete k-partite graph.

2. Let v1, v2, . . . , vk be k distinct vertices of a k-connected graph G. Let H be the graph formed from G by adding a new vertex w of degree k that is adjacent to each of v1, v2, . . . , vk. Show that κ(H) = k.

3. (a) Prove that if G is a k-connected graph, then G ∨ K₁ is (k + 1)-connected.

(b) Prove that if G is a k-edge-connected graph, then G ∨ K1is (k + 1)-edge-connected.

4. Let G be a graph with degree sequence d₁, d₂, . . . , d_n where d₁ ≥ d2 ≥

· · · ≥ d_n. Determine λ(G ∨ K₁).

5. Show for every k-connected graph G and every tree T of order k + 1 that there exists a subgraph of G isomorphic to T .

6. (a) Let G be a noncomplete graph of order n and connectivity k such that deg v ≥ (n + 2k − 2)/3 for every vertex v of G. Show that if S is a minimum vertex-cut of G, then G − S has exactly two components.

(b) Let G be a noncomplete graph of order n and connectivity k such that deg v ≥ (n + kt − t)/(t + 1) for some integer t ≥ 2. Show that if S is a vertex-cut of cardinality κ(G), then G − S has at most t components.

7. For a graph G of order n ≥ 2, define the k-connectivity κ_k(G) of G (2 ≤ k ≤ n) as the minimum number of vertices whose removal from G results in a graph with at least k components or a graph of order less than k. (Therefore, κ2(G) = κ(G).) A graph G is defined to be (`, k)-connected if κk(G) ≥ `. Let G be a graph of order n containing a set of at least k pairwise nonadjacent vertices. Show that if

deg_Gv ≥ n + (k − 1)(` − 2) k



for every v ∈ V (G), then G is (`, k)-connected.

8. Verify that Theorem 4.1 is best possible by showing that for each positive integer k, there exists a graph G of order n ≥ k + 1 such that δ(G) =

_n+k−3

2  and κ(G) < k.

9. Let a, b and c be positive integers with a ≤ b ≤ c. Prove that there exists a graph G with κ(G) = a, λ(G) = b and δ(G) = c.

EXERCISES FOR CHAPTER 4 111 10. The connection number con(G) of a connected graph G of order n ≥ 2 is the smallest integer k with 2 ≤ k ≤ n such that every induced subgraph of order k in G is connected. State and prove a theorem that gives a relationship between κ(G) and con(G) for a graph G of order n.

11. For an even integer k ≥ 2, show that the minimum size of a k-connected graph of order n is kn/2.

12. Prove or disprove: Let G be a nontrivial graph. For every vertex v of G, κ(G − v) = κ(G) or κ(G − v) = κ(G) − 1.

13. (a) Prove that if G is a k-connected graph and e is an edge of G, then G − e is (k − 1)-connected.

(b) Prove that if G is a k-edge-connected graph and e is an edge of G, then G − e is (k − 1)-edge-connected.

14. (a) Show that if G is a 0-regular graph, then κ(G) = λ(G).

(b) Show that if G is a 1-regular graph, then κ(G) = λ(G).

(c) Show that if G is a 2-regular graph, then κ(G) = λ(G).

(d) By (a) – (c) and Theorem 4.6, if G is r-regular, where 0 ≤ r ≤ 3, then κ(G) = λ(G). Find the minimum positive integer r for which there exists an r-regular graph G such that κ(G) 6= λ(G).

(e) Find the minimum positive integer r for which there exists an r-regular graph G such that λ(G) ≥ κ(G) + 2.

15. For a graph G, define κ(G) = max{κ(H)} and λ(G) = max{λ(H)}, where each maximum is taken over all subgraphs H of G. How are κ(G) and λ(G) related to κ(G) and λ(G), respectively, and to each other?

16. Let G1 and G2 be two k-connected graphs, where k ≥ 2, and let G be the set of all graphs obtained by adding k edges between G1 and G2. Determine max{κ(G) : G ∈ G}.

Section 4.2. Theorems of Menger and Whitney

17. Prove Corollary 4.12: Let G be a k-connected graph, k ≥ 1, and let S be any set of k vertices of G. If a graph H is obtained from G by adding a new vertex w and joining w to the vertices of S, then H is also k-connected.

18. Prove Corollary 4.13: If G is a k-connected graph, k ≥ 2, and u, v1, v2, . . ., vtare t + 1 distinct vertices of G, where 2 ≤ t ≤ k, then G contains a u − vi path for each i (1 ≤ i ≤ t), every two paths of which have only u in common.

19. Prove Corollary 4.14: A graph G of order n ≥ 2k is k-connected if and only if for every two disjoint sets V₁ and V₂ of k distinct vertices each, there exist k pairwise disjoint paths connecting V₁ and V₂.

20. Let G be a k-connected graph and let v be a vertex of G. For a positive integer t, define Gt to be the graph obtained from G by adding t new vertices u1, u2, . . . , utand all edges of the form uiw, where 1 ≤ i ≤ t and for which vw ∈ E(G). Show that Gtis k-connected.

21. Show that if G is a k-connected graph with nonempty disjoint subsets S₁ and S₂of V (G), then there exist k internally disjoint paths P₁, P₂, . . . , P_k such that each path P_i is a u − v path for some u ∈ S₁ and some v ∈ S₂ for i = 1, 2, . . . , k and |S₁∩ V (P_i)| = |S₂∩ V (P_i)| = 1.

22. Let G be a k-connected graph, k ≥ 3, and let v, v1, v2, . . . , vk−1 be k vertices of G. Show that G has a cycle containing all of v1, v2, . . . , vk−1

but not v and k − 1 internally disjoint v − ui paths Pi (1 ≤ i ≤ k − 1) such that for each i, the vertex ui is the only vertex of Pi on C.

23. Prove Theorem 4.18: A nontrivial graph G is k-edge-connected if and only if G contains k pairwise edge-disjoint u − v paths for each pair u, v of distinct vertices of G.

24. Prove or disprove: If G is a k-edge-connected graph and v, v₁, v₂, . . . , v_k are k + 1 vertices of G, then for i = 1, 2, . . . , k, there exist v − v_i paths P_i such that each path P_i contains exactly one vertex of {v₁, v₂, . . . , v_k}, namely v_i, and for i 6= j, P_i and P_j are edge-disjoint.

25. Prove or disprove: If G is a k-edge-connected graph with nonempty dis-joint subsets S1 and S2 of V (G), then there exist k edge-disjoint paths P1, P2, . . . , Pksuch that for each i, Pi is a u − v path for some u ∈ S1and some v ∈ S2 for i = 1, 2, . . . , k and |S1∩ V (Pi)| = |S2∩ V (Pi)| = 1.

26. Show that κ(Qn) = λ(Qn) = n for all positive integer n.

27. Assume that G is a graph in the proof of Theorem 4.17. Does the proof go through? If not, where does it fail?

28. Let G be a graph of order n with κ(G) ≥ 1. Prove that n ≥ κ(G)[diam(G) − 1] + 2.

29. A chorded cycle is a cycle C (of length at least 4) together with an edge that joins two nonconsecutive vertices of C. Prove that every 3-connected graph contains a chorded cycle but that this need not be the case for a 2-connected graph.

30. (a) Show that for every two vertices u and v of a 3-connected graph G, there exist two internally disjoint u − v paths of different lengths in G.

(b) Show that the result in (a) is not true in general if G is 2-connected.

EXERCISES FOR CHAPTER 4 113 31. (a) Prove that if G is a connected graph of order n and k is an integer with 1 ≤ k ≤ n − 3 such that the maximum number of internally disjoint u−v paths in G is k for every pair u, v of nonadjacent vertices of G, then G contains a vertex-cut with exactly k + 1 vertices.

(b) Let G be a k-connected graph of diameter k, where k ≥ 2. Prove that G contains k+1 distinct vertices v, v1, v2, . . . , vkand k internally disjoint v − vi paths Pi (1 ≤ i ≤ k) such that Pi has length i.

32. By Theorem 2.3, if G is a nontrivial graph of order n such that deg u + deg v ≥ n − 1 for every two nonadjacent vertices u and v of G, then G is connected, that is, G is 1-connected. Also, by Exercise 12 in Chapter 3, if G is a graph of order n ≥ 3 such that deg u + deg v ≥ n for every two nonadjacent vertices u and v of G, then G is nonseparable, that is, G is 2-connected. Consequently, for k = 1, 2 and k < n, if G is a graph of order n such that deg u + deg v ≥ n + k − 2 for every two nonadjacent vertices u and v of G, then G is k-connected. Prove that this is true for every positive integer k.

Chapter 5