Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned...

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V₁ and V₂, such that every edge joins a vertex in V₁ with a vertex in V₂. This means no edges are within V₁ or V₂ (or symbolically: ∀u, v ∈ V₁, {u,v} ∉ E and ∀u,v ∈ V₂, {u,v} ∉ E).

8(a) Show that the complete graph K₂ is a bipartite graph.

8(b) Prove that no complete graph K_n, where n > 2, is a bipartite graph.

8(c) Prove that every rooted tree forms a bipartite graph.

Expert Solution

Colby Messinger answered 2 years ago

Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices...

Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices u, v are called twins if the set of neighbors of u equals the set of neighbors of v (triplets, quadruplets etc are defined similarly). Show that G has twins. Bonus: Show that G has triplets. What about quadruplets, etc.?

Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E...

Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E is the set of edges, and w : E → R + is the weight of the edges (R + is the set of real positive numbers). Suppose T(G) is the set of all minimum spanning trees of G and is non-empty. If we know that the weight function w is a injection, i.e., no two edges in G have the same weight, then:...

Let G be a graph whose vertices are the integers 1 through 8, and let the...

Let G be a graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: vertex adjacent vertices 1 (2, 3, 4) 2 (1, 3, 4) 3 (1, 2, 4) 4 (1, 2, 3, 6) 5 (6, 7, 8) 6 (4, 5, 7) 7 (5, 6, 8) 8 (5, 7) Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the...

2. Let G be a bipartite graph with 10^7 left vertices and 20 right vertices. Two...

2. Let G be a bipartite graph with 10^7 left vertices and 20 right vertices. Two vertices u, v are called twins if the set of neighbors of u equals the set of neighbors of v (triplets, quadruplets etc are defined similarly). Show that G has twins. Show that G has triplets. What about quadruplets, etc.? 3. Show that there exists a bipartite graph with 10^5 left vertices and 20 right vertices without any twins. 4. Show that any graph...

. Provide a weighted directed graph G = (V, E, c) that includes three vertices a,...

. Provide a weighted directed graph G = (V, E, c) that includes three vertices a, b, and c, and for which the maximum-cost simple path P from a to b includes vertex c, but the subpath from a to c is not the maximum-cost path from a to c

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G),...

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G), u =v such that there is a trail from u to v which goes over every edge of G. The following sequence of questions is towards a proof of the following: Theorem 1. A connected graph G is semi-Eulerian (but not Eulerian) if and only if it has exactly two vertices of odd degree. Let G be semi-Eulerian with a trail t starting at...

Given an undirected graph G = (V,E), consisting of n vertices and m edges, with each...

Given an undirected graph G = (V,E), consisting of n vertices and m edges, with each edge labeled from the set {0,1}. Describe and analyze the worst-case time complexity of an efficient algorithm to find any cycle consisting of edges whose labels alternate 0,1.

If G = (V, E) is a graph and x ∈ V , let G \...

If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.

You are given a directed graph G(V,E) with n vertices and m edges. Let S be...

You are given a directed graph G(V,E) with n vertices and m edges. Let S be the subset of vertices in G that are able to reach some cycle in G. Design an O(n + m) time algorithm to compute the set S. You can assume that G is given to you in the adjacency-list representation.

Problem for submission: For which positive integers k can a simple graph G = (V, E)...

Problem for submission: For which positive integers k can a simple graph G = (V, E) be constructed such that: G has k vertexes, that is, |V | = k, G is bipartite, and its complement G is bipartite? Prove your answer is correct Please show and explain your full proof.

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