MELBOURNE, Australia (AP) -- Serena Williams made a bright start to her Australian Open title defense, holding up one finger and twirling in her neon yellow two-piece outfit to mark victory No. 1 after advancing 6-4, 7-5 over Camila Giorgi on Monday.

Williams withdrew from a warmup tournament because of inflammation in her left knee, so the win over Giorgi was her first completed match since her bid for a calendar-year Grand Slam ended in a semifinal loss at the U.S. Open.

The 21-time major winner was troubled initially by the bright sun while serving from one end of Rod Laver Arena, but didn’t show too many signs otherwise of discomfort as she beat the No. 34-ranked Giorgi in a challenging opener.

She dropped one service game in the match, while Giorgi struggled with 12 double-faults, including one on the decisive breakpoint in the 11th game of the second set.

Williams fired two aces and a service winner to finish it off.

“It wasn’t very easy -- she obviously plays really well ... (and) I haven’t played a competitive match in a really long time, but I think it was really good for me to come out here,” Williams said. “I was able to stay in it and stay calm today and I think that’s what matters most.”

Two-time Wimbledon champion Petra Kvitova earned the distinction of winning the first completed match at the tournament when she beat Thai qualifier Luksika Kumkhum earlier on Rod Laver Arena, avenging her upset three-set loss in the first round here in 2014.

Kvitova, who had to withdraw from a warmup tournament in China because of a stomach virus, said her preparation was disrupted and she was nervous ahead of the rematch with Kumkhum.

“Now I can breathe a little bit normally,” she said. “I got really nervous before the match. I didn’t sleep well actually. The memories from two years ago were a little bit difficult.”

No. 10 Carla Suarez Navarro advanced in straight sets, while No. 17 Sara Errani was the first of the seeded women eliminated with a 1-6, 7-5, 6-1 loss to Margarita Gasparyan. No. 24 Sloane Stephens, who beat Serena Williams in the 2013 Australian Open quarterfinals, went out in a 6-3, 6-3 loss to Chinese qualifier Wang Qiang.

No. 7 Kei Nishikori was the first of the seeded men’s players to advance, beating Philipp Kohlschreiber 6-4, 6-3, 6-3.

No. 15 David Goffin had a 3-6, 6-3, 6-4, 6-4 win over Sergiy Stakhovsky, and No. 22 Ivo Karlovic was the first of the seeded players to exit the tournament when he retired due to a left knee injury in the third set while trailing Federico Delbonis 7-6 (4), 6-4, 2-1.

The hours before the tournament began were overshadowed by reports alleging a number of players were suspected of being involved in match-fixing but had been allowed to continue playing.

The ATP, which runs the men’s tour, and Nigel Willerton, head of the Tennis Integrity Unit, represented the four governing bodies of tennis -- the ATP, the WTA, the Grand Slam Board and the International Tennis Federation -- in a joint news conference to reject the allegations published by the BBC and BuzzFeed News.

The reports said secret files exposed evidence of widespread suspected match-fixing at the top level of world tennis. The report said that in the last decade 16 players -- all at some stage ranked in the top 50 -- have been repeatedly flagged to the integrity unit.

The reports say the players -- it doesn’t identify them -- were allowed to continue competing.

ATP chairman Chris Kermode said tennis authorities “absolutely reject any suggestion that evidence of match-fixing has been suppressed for any reason, or isn’t being investigated.”

“We have stringent procedures and sanctions in place to deal with any suspected corruption and have shown we will act decisively when our integrity rules are broken,” Kermode said. “No player or official is immune ... Investigations follow where evidence leads.”

Tennis Integrity Unit investigations have resulted in 18 successful disciplinary cases, including five players and one official who have been banned from the sport for life.