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“As with any write-in campaign, the burden of execution rests with the candidate whose name is not on the ballot,” Mr. Miller told supporters. “At this point, without a single write-in ballot counted, Lisa Murkowski has no claim on a victory.”

Out-of-state election monitors are expected to arrive soon in Alaska for the ballot count that begins Wednesday. The deadline for a recount request is Dec. 4.

In Minnesota, Democrat Mark Dayton, a former U.S. senator, clung to a lead over Republican Tom Emmer of less than 1 percentage point in the three-way race, close enough likely to trigger a recount.

In the House, Republicans have already gained a net of at least 60 seats and could increase their new majority, depending on how the final handful of undecided races break.

Republican challengers hold the lead over Democratic incumbents in at least three seats in California, Illinois and Texas.

In Texas’ 27th District, for example, GOP challenger Blake Farenthold has been declared the winner, but 14-term incumbent Democratic Rep. Solomon P. Ortiz said election officials have discovered a bag of uncounted votes that could change the outcome.