A bipartite graph is drawn on a channel if the vertices of one partite set are...

A bipartite graph is drawn on a channel if the vertices of one partite set are placed on one line in the plane (in some order) and the vertices of the other partite set are placed on a line parallel to it and the edges are drawn as straight-line segments between them. Prove that a connected graph G can be drawn on a channel without edge crossings if and only if G is a caterpillar. (***Please do on paper)

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Colby Messinger answered 1 year 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.?

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...

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,...

Question 2: A bipartite graph with 2n vertices (namely |V1| = |V2| = n) is d-regular...

Question 2: A bipartite graph with 2n vertices (namely |V1| = |V2| = n) is d-regular if and only if the degree of every vertex in V1 ∪ V2 is exactly d. Show that a d-regular bipartite graph always has a perfect matching (a matching of size n that includes all vertices). ***Remarks: All the graphs here are without self loops and parallel or anti-parallel edges. A network is a directed graph with source s and sink t and capacity...

Network Structures Consider the set-up for the bipartite graph auction, with an equal number of buyers...

Network Structures Consider the set-up for the bipartite graph auction, with an equal number of buyers and sellers, and with each buyer having a valuation for the object being sold by each seller. Suppose that we have an instance of this problem in which there is a particular seller i who is the favorite: every buyer j has a strictly higher valuation for seller i’s object than for the object being sold by any other seller k. (In notation, we...

let G be a simple graph. show that the relation R on the set of vertices...

let G be a simple graph. show that the relation R on the set of vertices of G such that URV if and only if there is an edge associated with (u,v) is a symmetric irreflexive relation on G

An m × n grid graph has m rows of n vertices with vertices closest to...

An m × n grid graph has m rows of n vertices with vertices closest to each other connected by an edge. Find the greatest length of any path in such a graph, and provide a brief explanation as to why it is maximum. You can assume m, n ≥ 2. Please provide an explanation without using Hamilton Graph Theory.

3. What are the necessary and sufficient conditions for a bipartite graph to have a perfect...

3. What are the necessary and sufficient conditions for a bipartite graph to have a perfect matching? Justify your answer. 4. Illustrate Lemma 3.1.21 using the Peterson graph.

Write down the chromatic polynomials of (i)the complete graph K7; (ii)the complete bipartite graph K1,6. In...

Write down the chromatic polynomials of (i)the complete graph K7; (ii)the complete bipartite graph K1,6. In how many ways can these graphs be coloured with ten colours?

An eulerian walk is a sequence of vertices in a graph such that every edge is...

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

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