Already got wrong Chatgpt answer Plz don't use chat gpt 1. Let (G, X, Y) be a bipartite graph. Prove that there exist a matching M of and a subset Uof X such that|M| ≥ |N(U)| + |X| − |U|.(Hint: Use König's theorem)

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Answered: 1. Let (G, X, Y) be a bipartite graph.…

Linear Algebra: A Modern Introduction

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ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

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Show 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? ...1
Essentials of DISCRETE MATHEMATICS
5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).
2. Suppose n ≥ 1 is an integer. Consider an (n + 1) x (n + 1) grid of integer points; i.e. points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a path from the point (0, 0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to (a +1, b)) or ‘up' (i.e. from (a, b) to (a, b+1)). (a) Draw all distinct Binomial Paths with 2n steps when n = = 2. (b) Write down a correspondence that relates the Binomial Paths with 2n steps to strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B₁, be the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length 2n consisting of exactly n 1s and n Os. Construct a bijection f: Bn Sn. (You don't have to prove that it is a bijection.)
5. Let G = (V, E) be a bipartite graph with partite sets X and Y. Suppose X = {x₁,x₂,...,xn} and Y = {y₁, y2,...,Ym}. Show that Σdc(x₁) = |E|. i=1 m (HInt: Think graph fairy). (Also, note that Σdc(y;) = |E| as well, but just prove the above). j=1
Essentials of DISCRETE MATHEMATICS Section 2.4 - Relations and Equivalences
graph theory part 1 ,2 and 3

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1. Let (G, X, Y) be a bipartite graph. Prove that there exist a matching M of and a subset U of X such that |M| ≥ |N(U)| + |X| − |U|. (Hint: Use König's theorem)