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UPDATE 7-Oil edges up in quiet holiday trade, focus on crude glut

* Prices bounce a bit after steep fall last week

* U.S. adds rigs for record 19th straight week

* Oil inventories have dipped, but remain near records (New throughout, updates prices, market activity and comments; new byline, changes dateline, previously LONDON)

May 29 (Reuters) - Oil prices rose slightly on Monday, paring steep losses from last week, but the market remained cautious as increases in U.S. drilling activity have undercut an OPEC-led push to tighten supply.

Trading was subdued due to public holidays in China, the United States and Britain. The market remains uncertain about whether the extension of output cuts by OPEC and other producing countries will be enough to support prices.

Brent crude futures were up 37 cents at $52.52 per barrel at 1554 GMT. Brent fell nearly 3 percent the previous week.

U.S. West Texas Intermediate (WTI) crude futures were 42 cents higher at $50.22 per barrel.

The Organization of the Petroleum Exporting Countries and some non-OPEC producers pledged last week to extend production cuts of around 1.8 million barrels per day (bpd) until March 2018.

An initial agreement, in place since January, would have expired in June.

"This is a little bit of a bounce back from last Thursday when we had a really heavy drop," said James L. Williams, an energy economist at WTRG Economics in Arkansas.

"But nine months of current production levels is not going to be enough to meet OPEC's goal" of balancing supply, which is limiting price gains.

Commerzbank analyst Carsten Fritsch called Monday's price moves little more than "intraday noise."

High compliance with the cuts so far was unlikely to last, he said, which should keep physical oil stockpiles near record levels. So far the output cuts have not lifted oil prices much beyond $50 per barrel.

While OPEC and other countries including Russia are trying to draw down inventories, crude production has surged in the United States <C-OUT-T-EIA>, up 10 percent since mid-2016. U.S. output of more than 9.3 million bpd is close to levels in major producers Russia and Saudi Arabia.

U.S. drillers have added rigs for 19 straight weeks, bringing the total 722, the highest since April 2015 and the longest run of additions on record, according to energy services firm Baker Hughes Inc.

Even if the rig count did not rise further, Goldman Sachs estimated U.S. output would increase by 785,000 bpd between the fourth quarter of 2016 and the fourth quarter of 2017 across the Permian, Eagle Ford, Bakken and Niobrara shale plays.

Analysts will watch whether the OPEC output cuts can reduce the global crude glut.

"It's going to be all about inventories and whether they fall as much as OPEC thinks," said Greg McKenna, chief market strategist at futures brokerage AxiTrader.

(Additional reporting by Libby George in London and Henning Gloystein; in Singapore; editing by Jane Merriman and David Gregorio)