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(3) Suppose that G is a bipartite graph such that G = G(A, B). Show that G contains an even cycle. (4) Suppose that A = {u₁, U₂, u,} and B = {v₁, ₁,,,} such that u,v, E E(G)Vi s. Show that AUB V(G).

(3) Suppose that G is a bipartite graph such that G = G(A, B). Show that G contains an even cycle. (4) Suppose that A = {u₁, U₂, u,} and B = {v₁, ₁,,,} such that u,v, E E(G)Vi s. Show that AUB V(G).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 48E
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graph theory part 3 and 4
Exercise 2. Let G be a graph and let a and be G such that d(a) = A(G) and d(b) = 8(G) > 1. Suppose that b is the unique vertex in G such that d(b) = 5(G). Define the following two sets: A = {vG: d(v) <d(u) = N(v)} B = {v e G: d(v) > d(u) u € N(v)} (1) Show that a A. (2) Show that A is a stable set. (3) Suppose that G is a bipartite graph such that G=G(A, B). Show that G contains an even cycle. **** (4) Suppose that A = {u₁, U₂, ..., u.} and B = {V₁, V₁, -, v,} such that u¡v; € E(G) V1 ≤ i ≤ s. Show that AUB‡V(G). Exercice 3.
Transcribed Image Text:Exercise 2. Let G be a graph and let a and be G such that d(a) = A(G) and d(b) = 8(G) > 1. Suppose that b is the unique vertex in G such that d(b) = 5(G). Define the following two sets: A = {vG: d(v) <d(u) = N(v)} B = {v e G: d(v) > d(u) u € N(v)} (1) Show that a A. (2) Show that A is a stable set. (3) Suppose that G is a bipartite graph such that G=G(A, B). Show that G contains an even cycle. **** (4) Suppose that A = {u₁, U₂, ..., u.} and B = {V₁, V₁, -, v,} such that u¡v; € E(G) V1 ≤ i ≤ s. Show that AUB‡V(G). Exercice 3.
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