BRATISLAVA (Reuters) - Tens of thousands of Slovaks demanded early elections on Friday in protests across the country after Prime Minister Robert Fico’s resignation failed to quell public anger following the murder of an investigative journalist.

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Jan Kuciak, 27, and his fiancee were found shot dead at home last month. His journalism and then murder have rekindled public anger over the government’s failure to root out graft and cronyism 14 years after Slovakia joined the European Union.

In a third consecutive day of demonstrations, the protesters rallied in some 35 towns and cities, waving signs that read “Government Doesn’t Work” and chanting “Enough of Fico”.

“Coalition leaders are clinging to power at any cost, to cover up all the corruption scandals,” rally organizer Karolina Farska told the crowd in the capital Bratislava, estimated at 50,000 by public broadcaster RTVS.

The death of Kuciak -- who reported on fraud allegations against businessmen with political ties -- has provoked Slovakia’s biggest protests since the fall of communism in 1989.

The long-serving Fico stepped aside on Thursday, handing the government reins to a deputy after his leftist Smer party and its coalition partners agreed a change of prime minister to keep the government going. His interior minister also quit this week. But critics say Fico will continue to hold sway.

“I am willing to protest every week as I feel we have a good head start towards change,” Tereza Klimcova, a 29-year-old lawyer, said at the Bratislava protest. “We can’t stop now or else the pressure will disappear, as happened with all previous scandals.”

Fico has vowed to remain an active party leader, stoking fears he will continue to wield influence over his close ally Peter Pellegrini who has been asked to form a new government. Pellegrini is expected to announce appointments next week.

“As Smer chairman, (Fico) will continue to tutor the new prime minister,” said Martin Slosiarik of the Focus polling agency.

Just before his death, Kuciak had been looking into suspected mafia links of Italians with businesses in Slovakia.

In his final report published after his death, Kuciak said one of these Italians had once been a business partner of two Slovaks who later worked in Fico’s office. Both have resigned but deny connections to the murder. The Italian businessman has denied any mafia connections.

Nobody has been charged in the killings.