Exercise 2. Let G be a graph and let a and b E 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) = 8(G). Define the following two sets: A = {ve G: d(v) < d(u) u € N(v)} B = {veG: 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 = {₁, ₂,..., us} and B = {v₁, V₁,..., Us} such that uv; E E(G) V s. Show that AUB V(G).
Exercise 2. Let G be a graph and let a and b E 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) = 8(G). Define the following two sets: A = {ve G: d(v) < d(u) u € N(v)} B = {veG: 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 = {₁, ₂,..., us} and B = {v₁, V₁,..., Us} such that uv; E E(G) V s. Show that AUB V(G).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Exercise 2.
Let G be a graph and let a and b € 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) = 8(G). Define the following two sets:
A = {ve G: d(v) < d(u) u € N(v)}
B = {ve G: d(v) > d(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, ₂, ..., u.} and B = {V₁, V₁,..., v.} such that u¡v; € E(G) V1 ≤ i ≤
s. Show that AUBV(G).
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