For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V...

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such that (V, E0 ) and (V, E00) are spanning trees which are not isomorphic.

Expert Solution

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such that (V, E0 ) and (V, E00) are spanning trees which are not isomorphic.

Colby Messinger answered 1 year ago

Let Kn denote the simple graph on n vertices. (a) Let v be some vertex of...

Let Kn denote the simple graph on n vertices. (a) Let v be some vertex of Kn and consider K n − v, the graph obtained by deleting v. Prove that K n − v is isomorphic to K n−1 . (b) Use mathematical induction on n to prove the following statement: K n , the complete graph on n vertices, has n(n-1)/2 edges

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 Kn,...

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.

Problem 2. Consider a graph G = (V,E) where |V|=n. 2(a) What is the total number...

Problem 2. Consider a graph G = (V,E) where |V|=n. 2(a) What is the total number of possible paths of length k ≥ 0 in G from a given starting vertex s and ending vertex t? Hint: a path of length k is a sequence of k + 1 vertices without duplicates. 2(b) What is the total number of possible paths of any length in G from a given starting vertex s and ending vertex t? 2(c) What is the...

Showing your work, determine the values of m and n for which the complete bipartite graph...

Showing your work, determine the values of m and n for which the complete bipartite graph Km,n has an a) Euler circuit? b) Euler path?

A.Please answer these questions: 1) For which n does the complete graph Kn have an Eulerian...

A.Please answer these questions: 1) For which n does the complete graph Kn have an Eulerian circuit? 2) For which n does the complete graph Kn have a Hamiltonian Cycle? 3）For which r, s does the complete bipartite graph Kr,s have an Eulerian circuit? 4） For which r, s does the complete bipartite graph Kr,s have a Hamiltonian Cycle? 5）Find a graph that has an Eulerian circuit but no Hamiltonian Cycle? 6）Find a graph that has a Hamiltonian Cycle but...

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.

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3...

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. a) Draw a maximal plane graphs on six vertices. b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a convex n-gon...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let G = (V, E) be a directed graph, with source s ∈ V, sink t...

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum s-t cut (i.e., an s-t of capacity strictly less than that of all other s-t cuts).

Question