= {V1, V2, V3,, Un}, 10. Let G = (V,E) be a loop-free connected undirected graph where V n ≥ 2, deg(v1) = 1, and deg(vi) ≥ 2 for 2≤ i ≤n. Prove that G must have a cycle.

1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEV

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A Hamiltonian cycle of an undirected graph G = (V, E) is a simple cycle that contains each vertex in Vonly once. A Hamiltonian path of an undirected graph G = (V, E) is a simple path that contains each vertexin V only once. Let Ham-Cycle = {<G, u, v> : there is a Hamiltonian cycle between u and v in graph G} andHam-Path = {< G, u, v>: there is a Hamiltonian path between u and v in graph G}. Prove that Ham-Path is NP-hard using the reduction technique, i.e. find a known NP-completeproblem L ≤p Ham-Path (reduce from the Ham-Cycle problem). Make sure to give the followingdetails: 1) Describe an algorithm to compute a function f mapping every instance of L to an instance of Ham-Path 2) Prove that x ∈ L if and only if f(x) ∈ Ham-Path 3) Show that f runs in polynomial time

Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.

Consider a connected, undirected graph G with n vertices and m edges. The graph G has a unique cycle of length k (3 <= k <= n). Prove that the graph G must contain at least k vertices of degree 2.

Let H be a planar graph with n vertices and m edges, where n >= 5.  Prove that if H does not have any cycles of length 3 and 4 then 3m <= 5n - 10.

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1. Prove that if v1 and v2 are distinct vertices of a graph G = (V,E) and a path exists in G from v1 to v2 , then there is a simple path in G from v1 to v2 .

Question 2: Let G be a planar graph on at least 4 vertices. Suppose that the minimum degree of G is 5. Show that the number of vertices of degree 5 in G is at least 12.