The minimum number of colours that is sufficient to vertex-colour any planar graph is _____________ .

In which one of the following page replacement algorithms it is possible for the page fault rate to increase even when the number of allocated frames increases?

Consider the following processes, with the arrival time and the length of the CPU burst given in milliseconds. The scheduling algorithm used is preemptive shortest remaining-time first.

Process	Arrival Time	Burst Time
P1 P2 P3 P4	0 3 7 8	10 6 1 3

The average turn around time of these processes is milliseconds.

Consider a non-negative counting semaphore $$S.$$ The operation $$P(S)$$ decrements $$S,$$ and $$V(S)$$ increments $$S.$$ During an execution, $$20$$ $$P(S)$$ operations and $$12$$ $$V(S)$$ operations are issued in some order. The largest initial value of $$S$$ for which at least one $$P(S)$$ operation will remain blocked is _____________ .