Let G be a connected planar graph of order n > 3 and has no cycles of length 3. Prove that q≤2n4, where q is the number of edges in G.
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- Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.Consider a graph G where the vertices are points (a, b) with integer coordinates, and {(ai, b;), (aj, b;)} is an edge whenever a; + aj = b; + b;. Show that G is bipartite if and only if it has at most two vertices on the line y = x.Construct a simple graph with vertices J,K,L,MJ,K,L,M whose degrees are 1, 0, 1, 0 respectively. List the edges separated by a comma?
- Q5. a) If G (V, E) is a connected undirected graph with |E| = 19 and deg(v) >=4 for all v ∈ V , what is the maximum value for |V|. b) What will be the number of edges in graph Q8 ? Explain with proper justification.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.Explain thoroughly!! And please complete all 4 parts
- Q₁. Construct a connected graph with at least 12 vertices such that the minimal dominating set has 6 vertices and the minimum dominating set has 4 vertices.Compute the vertex set V and edge set E of the graph G = (V,E) whose Prufer Code is (3, 3, 3,3, 3, 3). Select one: OV={1,2,...,8} and E = {(1,3), (2, 4), (4, 3), (5, 3), (6, 3), (7, 3), (8, 7)} {(1,4), (2, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 3)} %3D OV={1,2,..., 8} and E = OV= {1,2,... ,8} and E = {(1, 4), (2, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 7)} OV= {1,2,..., 8} and E = {(1,3), (2, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 3)} Clear my choiceWhere are the vertices?
- 2.) Determine how many backtracks are required to use every edge at least once returning to where you started. Indicate where the backtracks occur.construct a graph with vertices Q,R,S,T whose degrees are 3,2,2,1 what is the edge set?Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6.Please write in complete sentences, include all details, showall of your work, and clarify all of your reasoning.