As for the first question, the answer is obviously no. You only have to look at the figures of unemployment in my country, Spain, to have a good example.
Regarding the second question, although more debatable, the answer would be no again. If so, the government would be responsible improve the matching process between employers and employees. But for all educational levels, not only for graduate students.
Your question is a bit more nuanced than a straight no.
If the government has sponsored the student in their studies they might have a greater responsibility, or at least an interest, in offering a position as a return on their investment.
Overall I am in favour of decoupling employment with university education as it can tend to narrow what is valued - so students only see as valuable that which tangibly makes them more employable rather than the untangibles which might not get them employment but which make them better to employ.
I agre with some of the earlier answers. Guranteeing employment for graduates is bad policy for many reasons. The example of Sri Lanka shows that when this is the case, the incentives for graduates to acuire skills is less. Also, this creates distortions in the labour market. Finally the competition for entering university increases, as this gurantees a job.