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1. For a graph G with vertex set V, and disjoint sets A, B CV, let e(A, B) be the number of edges of G with one end in A and the other end in B, and let d(A) e(A, VA). Prove, for any X, Y CV, that d(X) + d(Y) ≥ d(XUY) + d(XNY). (Hint: one option is to show that d(X)+d(Y) = d(XUY)+d(XÑY)+2e(X\Y, Y\X).)

1. For a graph G with vertex set V, and disjoint sets A, B CV, let e(A, B) be the number of edges of G with one end in A and the other end in B, and let d(A) e(A, VA). Prove, for any X, Y CV, that d(X) + d(Y) ≥ d(XUY) + d(XNY). (Hint: one option is to show that d(X)+d(Y) = d(XUY)+d(XÑY)+2e(X\Y, Y\X).)

Advanced Engineering Mathematics
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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I have to prove this as an exercise for my Graph Theory course but I don't really understand what this means. Can someone please give a hint or an idea of how I could approach/prove this proof? Thank you very much!!

1. For a graph G with vertex set V, and disjoint sets A, B C V, let e(A, B) be the number of edges of G with one end in A and the other end in B, and let d(A) e(A, V \ A). Prove, for any X, Y CV, that d(X) + d(Y) ≥ d(XUY) + d(XNY). (Hint: one option is to show that d(X)+d(Y) = d(XUY)+d(XÑY)+2e(X\Y, Y\X).) =
Transcribed Image Text:1. For a graph G with vertex set V, and disjoint sets A, B C V, let e(A, B) be the number of edges of G with one end in A and the other end in B, and let d(A) e(A, V \ A). Prove, for any X, Y CV, that d(X) + d(Y) ≥ d(XUY) + d(XNY). (Hint: one option is to show that d(X)+d(Y) = d(XUY)+d(XÑY)+2e(X\Y, Y\X).) =
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