June 28 (Reuters) - Euro zone government bond yields rose on Tuesday, with investors focusing on inflation risks, while spreads between core and periphery countries tightened as the European Central Bank reiterated its anti-fragmentation tool should not have limits.

The ECB announced in mid-June plans to tackle fragmentation in euro zone bond markets, or an excessive widening of spreads between core and peripheral yields that might hamper monetary policy transmission across the bloc.

ECB policymaker Pierre Wunsch said on Tuesday the bank should offer unlimited support to euro zone members facing an unjustified surge in borrowing costs.

ECB president Christine Lagarde said it would act decisively on any deterioration in medium-term inflation prospects, while reining in any disorderly widening of spreads.

The ECB must balance the need to tame inflation against the risk of triggering a recession.

Analysts say there are too many concerns about inflation and not enough about a potential recession to bring the bear market for bonds to a close, while waiting for euro zone inflation data later this week.

Oil rose $1.9 a barrel as Group of Seven nations promised a potential price cap on Russian oil and gas that would hit Russian President Vladimir Putin's war chest.

Germany's 10-year government bond yield, the benchmark in the euro zone, rose 10 basis points (bps) to 1.64%, after hitting an almost one-week high at 1.673%.

"This is potentially an interesting week as we will see if central bankers intend to moderate their hawkish tones while focusing also on growth risks," said Andrew Mulliner, head of Global Aggregate Strategies at Janus Henderson, mentioning this week's inflation data.

German will release consumer price numbers on Wednesday, while euro area data is due on Friday.

Several policymakers will speak at the ECB Forum on Central Banking 2022 in Sintra, Portugal, on Tuesday and Wednesday.

New York Federal Reserve Bank President John Williams called for further rapid U.S. interest rate hikes to slow inflation, including a possible second 75-basis point rate hike next month, but said he did not expect a U.S. recession.

"A credible anti-fragmentation tool would enable the ECB to steepen its monetary tightening path, and then we would expect yields to rise," said Hetal Mehta, senior European economist at Legal and General Investment Management.

"But we also have to consider that the market pricing of future rate hikes is quite aggressive given recession risks," she added.

Italy's 10-year government bond yield rose 4 bps to 3.67%, with the spread between Italian and German 10-year yields tightening to 201 bps.

Janus Henderson's Mulliner said he expected the spread to remain around the current levels before the ECB's July policy meeting as markets wanted to see how credible the anti-fragmentation tool was before taking any price action.

The ECB will likely drain cash from the banking system to offset any bond purchases made to cap borrowing costs for indebted euro zone states, sources told Reuters.

"Periphery spreads could come under pressure from the latest ECB sources, pouring cold water on the idea that the ECB could sell Bunds to sterilise periphery purchases under the new anti-fragmentation tool," Commerzbank analysts said.

(Reporting by Stefano Rebaudo Editing by Mark Potter)