let G be an n vertex simple graph other than Kn Prove that if G is not k-connected, than G has a separating set of size k-1
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- Let G be a graph with n ≥ 3 vertices that has a clique of size n − 2 but no cliques of size n − 1.Prove that G has two distinct independent sets of size 2. Show your work and complete proof.Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible number of edges in G? Show that this bound is sharp.A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?
- 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.I want this to be considered as a Advanced Math question pls. . Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it has two sets of vertices, with 3 vertices in one set and 4 in the other. Every vertex in one set is connected to every vertex in the other set, but there are no connections within a set. Calculate the number of edges in graph G. Also, determine if the graph G contains an Euler path or circuit, and justify your answer.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.
- Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does not necessarily contain a (u, v)-path Q internally-disjoint from PShow 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? ...1Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and let m be the minimum degree of the vertices of G. Show that 2e/v≥m
- 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of degree 1 or an isolated vertex.let n>=2 be a natural number Let V be the set of people in a party of n people. Use set-builder notation o define E in the undirected graph G=(V,E) where there is an edge between two vertices u and v if the person u and v have met each other. What does the degree of edge in G represent? Prove that there cannot be two vertices u and v in V such that deg(u)=0 and deg(v)=n-1 at the same time. Use part 2 and 3 along with pigeon hole principle to show that in every party of n people, there is always two people who both have met exactly the same number of people.Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced subgraph isomorphic to k1,r+1 has a 1-factor. Show that this is not true if you replace r connected by r edge connected