The lawsuit filed Friday challenges a House map drawn by Republicans in 2016 after some districts were found to be racially gerrymandered. Democrats claim that House districts have been crafted to assure Republican dominance since party leaders first drew them in 2011. The suit argues that the map violates State Constitution clauses guaranteeing free elections, freedom of speech and assembly, and equal protection under the law.

If the suit succeeds, it could open the way for Democratic gains in North Carolina. After the Pennsylvania Supreme Court struck down a Republican gerrymander of that state’s congressional map last year, Democrats captured nine of 18 House seats, a gain of four from 2016.

North Carolina’s Republican leaders have long said that they drew the state’s congressional map for maximum political gain. One of the principal drafters of the most recent version of the House boundaries boasted in 2016 that he had given Republicans a 10-to-3 edge in seats “because I do not believe it’s possible to draw a map with 11 Republicans and two Democrats.” But Republicans say Democrats gerrymandered the state in their direction for decades when they controlled the Legislature.

Since the House map was first drafted in 2011, it has become an exercise in political recidivism — challenged often in court, struck down three times, but always emerging with a pronounced bias favoring Republicans.

Beginning in 2013, critics charged in state and later federal courts that two of the 13 House districts were racial gerrymanders. The state court disagreed, but the United States Supreme Court upheld the complaints and ordered the districts redrawn in 2016.

Republican leaders not only complied, but also said that the new maps would avoid any accusations of racial bias by being drawn strictly for political gain. That led critics to sue again, charging this time that the entire House map was a partisan gerrymander. A three-judge federal panel struck the map down on those grounds two times before the Supreme Court reversed the panel in June.

Separately on Friday, the former chairman of the North Carolina Republican Party, Robin Hayes, agreed to plead guilty to federal charges of lying to the F.B.I. about a scheme to influence the state insurance commissioner. Mr. Hayes and three other men were indicted in April on conspiracy and bribery charges stemming from what prosecutors say was an effort to win favorable treatment for an insurance company controlled by one of the men.