Hi,
As an EU graduate there should not be any problems with staying in Denmark. However you need a EU residence document.
As an EU national you are free to enter Denmark and reside in the country for up to three months without a residence document.
If you are looking for work, you will be able to reside in Denmark for up to six months without a residence document.
If your grounds for residence are terminated – e.g. if you stop working or studying – you must apply for a new residence document on other grounds – e.g. as a person with sufficient funds.
In relation to having grounds for residence under EU rules, being a person with sufficient funds means that you are self-supporting and thus have access to funds or income that can be used to maintain yourself and your family to such a degree that it can be assumed that you will not become a public burden.
If you have a residence document, you may remain in Denmark indefinitely, provided you meet the conditions for your grounds for residence. There is no date of expiry for residence documents.
You can read more about this on the website of SIRI (” The Danish Agency for International Recruitment and Integration”). This is also to SIRI that you must apply for the residence document.
Read more about the EU residence document here: https://www.nyidanmark.dk/en-GB/You-want-to-apply/Residence-as-a-Nordic-citizen-or-EU-or-EEA-citizen.
And you are allowed to receive Dagpenge, after the same rules/requirements as danish citizens.
So the answer is: you can stay for more than 6 months, but you need to apply for the residence document (not residence permit).
And you can continue to receive Dagpenge for up to 2 years from your graduation date.
Please let me know, if you have further comments.
Best regards,
Anders Weber, http://www.a-kasser.dk