4. Prove that for each n€ Z+ there exists a loop-free connected undirected graph G = (V,E), where |V| 2n and which has two vertices of degree i for every 1≤i≤n. =
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- 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? ...110. Let G = (V, E) be a loop-free connected planar graph. If G is isomorphic to its dual and|V | = n, what is |E|?Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.
- 3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.10. Let G = (V, E) be a loop-free connected undirected graph where V = {v1, v2, v3, . . . , vn},n ≥ 2, deg(v1) = 1, and deg(vi) ≥ 2 for 2 ≤ i ≤ n. Prove that G must have a cycle.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).
- 13. Let G = (V, E) be an undirected connected loop-free planar graph. Suppose G determines53 regions. If, for some planar embedding of G, each region has at least five edges in its boundary,prove that |V | > 81.In chess. A “knight’s move” consists of two squares either vertically or horizontally and then one square is a perpendicular direction. Depending on where the knight is situated, he has a minimum mobility of two moves—when in a corner—and a maximum mobility of eight moves. Let C be a graph with v=64, its vertices corresponding to the squares of a chessboard. Let two vertices of C be joined by an edge whenever a knight can go from one of the corresponding squares to the other in one move. Does C have an Euler Walk? Explain, but you do not have to draw C to answer.Let G be a connected graph that has exactly 4 vertices of odd degree: v1, V2, V3 and V4. Show that there are paths with no repeated edges from v1 to V2, and from v3 to v4, such that every edge in G is in exactly one of these paths.
- 9. The connected undirected graph G = (V, E) has 30 edges. What is the maximum value that|V | can have?14. Let G be the graph with vertices v1, v2, and v3 as shown below. What is the number of walks of length 3 from v2 to v3? v2 V34. Let n ∈ Z+ with n ≥ 4, and let the vertex set V ′ for the complete graph Kn−1 be{v1, v2, v3, . . . , vn−1}. Now construct the loop-free undirected graph Gn = (V, E) from Kn−1 asfollows: V = V ′ ∪ {v}, and E consists of all the edges in Kn−1 except for the edge {v1, v2}, whichis replaced by the pair of edges {v1, v} and {v, v2}.a) Determine deg(x) + deg(y) for all nonadjacent vertices x and y in V .b) Does Gn have a Hamilton cycle?c) How large is the edge set E?d) Do the results in parts (b) and (c) contradict Corollary 11.6?
