2. Prove the following: For any simple graph G = (V. E), V can be partitioned into X, Y, Z such that E(X,Y,Z) ≥ 2E\/3. Here E(X, Y, Z) is the set of edges between X, Y, Z. In other words, E(X,Y,Z) = {e E E: each of X, Y, Z may contain at most one end of e}; and "X, Y, Z is a partition of V" means: XUYUZ = V and XnY=XnZ=Ynz=0.
2. Prove the following: For any simple graph G = (V. E), V can be partitioned into X, Y, Z such that E(X,Y,Z) ≥ 2E\/3. Here E(X, Y, Z) is the set of edges between X, Y, Z. In other words, E(X,Y,Z) = {e E E: each of X, Y, Z may contain at most one end of e}; and "X, Y, Z is a partition of V" means: XUYUZ = V and XnY=XnZ=Ynz=0.
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:2. Prove the following:
For any simple graph G = (V, E), V can be partitioned into X, Y, Z such that
|E(X,Y,Z) ≥ 2|E|/3.
Here E(X, Y, Z) is the set of edges between X, Y, Z. In other words,
E(X,Y,Z) = {e € E: each of X, Y, Z may contain at most one end of e};
and "X, Y, Z is a partition of V" means:
XUYUZ = V and XnY=XnZ=Ynz=0.
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