Ouestion 3 (a) Suppose that T is a tree with n vertices, two of which have degrees r and s, respectively, where r2s2 2. Prove that T has at least r+s-2 vertices of degree one.
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- prove that the degree of a regular complete tripartite graph kr,s,t with n vertices is given by 2n/3Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that if G is not k-connected, then G contains a vertex-cut U with |U| = k − 1? b) if G is not k-edge-connected, then G contains an edge-cut X with |X| = k − 1Prove that every connected planar graph with less than 12 vertices has a vertex of degree at most 4. [Hint: Assume that every vertex has degree at least 5 to obtain a lower bound on e (together with the upper bound on e in the corollary) that implies v ≥ 12.]
- (b) Suppose G is a simple connected graph with 12 vertices and 16 edges. Suppose 4 of its vertices are degree 1, and 3 of its vertices are degree 2. Prove that G is planar. (Hint: Kuratowski) (c) Let G be any simple connected planar graph with n vertices and e edges. Suppose there are exactly y vertices of degree 2. Assume that n - y > 3. Prove that e < 3n - y - 6. (Hint: Explain why the degree-2 vertices can be erased, and how to take care of any resulting loops or multiple edges.) (d) Suppose that a connected simple graph G' has exactly 10 vertices of degree 4, 8 vertices of degree 5, and all other vertices have degree 7. Find the maximum possible number of degree-7 vertices G could have, so that G would still be planar.Prove that for every odd n ≥ 5 there exist a graph of n+1 vertices n of which have degree 3, and the last one has degree n not equal 3Determine the largest positive integer k such that χ(H) = χ(G) = k, where H is obtained from a nonempty graph G by subdividing each edge of G exactly once.
- 3. (b) The degree of every vertex of a graph G is one of three consecutive integers. If, for each of the three consecutive integers x, the graph G contains exactly x vertices of degree x, prove that two-thirds of the vertices of G have odd degree. (c) Construct a simple graph with 12 vertices satisfying the property described in part (b).Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.A certain graph G of order n and size m = 17 has one vertex of degree 1, twovertices of degree 2, one vertex of degree 3, and two vertices of degree 5. The remaining vertices of G have degree 4. What is n?