Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 20H / b / iii

Table of contents
- Solution iii

Solution

The maximum flow is not unique. For any

x \in [1, 2]

f (S a) = f (a d) = 1, f (S b) = 3, f (S c) = f (ce) = 4,

and split the flow from

b

by

f (b d) = x, f (b e) = 3 - x .

Flow conservation then requires

f (d T) = 1 + x, f (e T) = 7 - x .

All capacities are respected precisely throughout the displayed interval, and every member of this one-parameter family has value

8

. For example,

x = 1

and

x = 2

give distinct maximum flows.

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