4.1.5. (-) Let G be a connected graph with at least three vertices. Form G' from G by adding an edge with endpoints x, y whenever de (x, y) = 2. Prove that G'is 2-connected.
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- Let H be any graph, and let H’ be the graph with V(H’) = V(H) U {v} for some vertex v not in V(H), and E(H’) = E(H). For each positive integer n and graph H, find ex(n, H’) in terms of ex(n, H).Prove (Menger) if x, y are vertices of a graph G and xy e E(G), then the minimum size of an x,y-cut equals the maximum number of pairwise internally disjoint x,y-paths6. (In this problem, y denotes the domination number of a graph.) Let G be a graph such that y(G) > 3. Prove that diam(G) < 2.
- Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph obtained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) 4e)}.6. A vertex v of G is essential if v is covered by every maximum matching in G. That is, a'(G – v) = a'(G) – 1. Describe an infinite family of graphs which contain no essential vertices.5. Let G = (V, E) be a graph with vertex-set V = {1,2,3,4} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2,4)}. (a) Draw the graph. Find (b) maximal degree, i.e. A(G), (c) minimal degree, i.e. 8(G), (d) the size of biggest clique, i.e. w(G), (e) the size of biggest independent set, i.e. a(G), ter (f) the minimal number of colours needed to color the graph, i.e. x(G).
- = 2 and let G₁ = (V, E) be the graph with vertex set V i+1}:1Let F, F' be forests on the same set of vertices, with ||F|| < ||F'||. Show that F' has an edge e such that F + e is again a forest.2.6.1. (a) Prove that graph G = (V, G) is even if and only if the edge induced subgraph G[E] is an even subgraph of G.Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)(i) Let e1, e2, . . . , one connection of the graph G and let xi−1 and xi be the sites of the connection ei (1 ≤ i ≤ n). Show that every closed walk (e1, e2, . . . , en) is of length at least 3 with pairs of distinct pointsx1, x2, . . . , xn cycles.(ii) A graph containing no cycles is called acyclic. A walk is acyclic if the subgraph consisting of points and links of the walk is acyclic. Prove: a walk has all distinct points if and only if it is an acyclic sequence.(iii) If and are in different points of the graph G and if there is a walk in G from u to v, show that then there is an acyclic sequence from u to v.(Explain precisely that every shortest walk from u to v is actually an acyclic path.)Show that For n > 1 let Gn be the simple graph with vertex set V(Gn) = {1,2, ., n} in which two different vertices i and j are adjacent whenever j is a multiple of i or i is a multiple of j. For what n is Gn planar? ...1SEE MORE QUESTIONS