British Airways has said it operated more flights than planned as cabin crew began a second five-day strike.
The airline said it had reintroduced all flights from Heathrow to New York's JFK airport and would add more.
It insisted its operations went "very well" during last week's action and that it had made another "good start".
The Unite union claimed BA cancelled more than 100 flights on Sunday and said its strikes continued to be strongly supported.
Unite has said it would suspend industrial action if BA restored travel perks it took from those cabin crew who went on strike in March.
The latest walkout coincides with the half-term holidays, and could be followed by more action starting on 5 June.
A BA spokesperson said: "Our global operations went very well throughout the first week of Unite's strike action and got off to another good start today.
"We have announced a larger schedule at Heathrow for this round of strikes, because of the numbers of crew reporting for work.
"We will continue to operate 100% of our schedule at Gatwick and London City airports.
"At Heathrow, we will operate to more than 70% of longhaul flights (up from more than 60% last week) and more than 55% of shorthaul flights (up from more than 50% last week)."
Negotiations ended without agreement on Friday, with Unite blaming BA's chief executive Willie Walsh for the deadlock.
Conciliation service Acas will be contacting both sides to arrange further talks.
BA says the strike has cost it about £85m - an average of £6m for each of the 15 strike days so far.
The dispute is set to dominate Unite's annual conference in Manchester, where about 700 delegates will gather on Monday.
The union's joint leader Derek Simpson said he was prepared to meet "anytime, anywhere" to continue discussions.
But he told the BBC Mr Walsh was "prepared to let the strike run without any attempt, casual or otherwise" to resolve it.
BA denied this, saying their chief executive had made it clear he was prepared to go to Manchester to continue negotiations.
Mr Simpson claimed the airline was being "unreasonable" by refusing to reinstate the travel allowances, saying this had caused a loss of confidence among workers.
"We've been doing all we can to build that confidence back but it's very difficult," he said.
"If we could get the confidence back and if we could get the travel allowances back, I believe a deal is doable.
"I think Willie Walsh believes a deal is do-able but he seems to be intent on regime change [within the stewards branch of the union], wanting to punish the workforce.
"He knows full well our people are not going to go back humiliated or punished after the dispute is settled."
BA staff returned to work for one day between the two strike periods, but the airline said that although the "vast majority" of flights were operating it was impossible to run a full service.
Unite claimed 102 BA flights were cancelled on Sunday - including 22 long-haul flights.
A BA spokeswoman said because most leisure flights fly out of Gatwick rather than Heathrow, the disruption to half term holidays would be limited.