The U.S. dollar tumbled Tuesday, as the deutsche mark romped and the French franc remained in the dog house.

In Europe, the French franc weakened, extending a slide begun late Monday on news that Fi- nance Minister Jean Arthuis had been implicated in an investigation into alleged unlawful political party financing.Adding to the French franc's woes, dealers said, has been the latest Paris bombing attack, continued worries on the budget front and generalized deutsche mark strength against all the major European currencies.

Dealers also cited a German press article saying that the Franco-German motor of European monetary union was sputtering because of the franc's weakness. The article, by German professor Alfred Grosser, said that while France and Germany both support a single currency, French and German interests actually clash in other important economic and social areas.

The French franc fell through key psychological support at the 3.5000 franc-per-mark level in Tuesday afternoon trading in Paris.

Dealers said they had not seen any intervention by the Bank of France, but they said the likelihood of intervention is high if the franc slips much further.

''The Bank of France has painted itself into a corner by cutting rates on Monday and signaling that the franc crisis is over," one Paris dealer said. "Because now they are faced with hiking rates again and that doesn't do much for market confidence," the dealer said.

Players again were establishing long positions in deutsche marks and Swiss francs in anticipation of another bout of European monetary turbulence.

Dealers said that while the French franc has moved on specific news, the market is beginning to focus more and more on the "sustainability" of the Bank of France's policy of supporting the franc through very high real short- term interest rates.

''When the market moves on the kind of silly news that we've seen in France recently, you know that there is some kind of underlying problem there," one Paris dealer said.

There are also signs of increasing political opposition to the current franc policy.

Philippe Seguin, the president of the French National Assembly, severely criticized recent Bank of France policy.