In light of the exit polling, it looks like the UK will have a hung Parliament. In the event that no coalition/working relationship deal is made, am I correct in thinking that the country is then run by a minority government? How does it work exactly?
A minority government can be formed if no party gets a majority. Should they be unable to put together a coalition, they can form a minority government and fill all the ministerial positions themselves.
This Independent article explains how minority governments are formed:
How can they exist?
If a proposed government wins a confidence vote in parliament, it becomes the government. Other parties could support a government without actually joining it.
This could happen for a variety of reasons: the supporting party might think the resulting government was a better option than something else, the minority government might be the only option to form a government other than another election, or the supporters might get concessions.
Basically, the party will still have to cooperate with smaller parties to pass legislation in Parliament as it requires a majority.
This BBC article explains this:
If the Conservatives or Labour are unable to put together a coalition or decide to go it alone, they can form a minority government, filling all the ministerial positions themselves.
This party would be unable to pass laws and legislation without the votes of other parties that are not part of the government.
It's worth noting that minority governments often don't last long as it's difficult to govern and pass legislation in Parliament. An notable one would be Harold Wilson's minority Labour government that lasted only for 7 months in 1974, though he managed to win a small majority after calling an election.
A relative more successful1 one would be John Major's government that became a minority after losses ay by-elections. It existed for 2 years from 1996 to 1997.
1 It's considered successful as it managed to last much longer than Wilson's minority government in 1974.