In what appears to be a showdown with the nation's number one fast-food burger chain, Roy Rogers will cut the prices of its hamburgers and cheeseburgers starting Tuesday. However, other leading burger chains say they will not follow suit.

Roy Rogers, a division of the Marriott Corp., will slice prices of its burgers 4 to 10 cents, depending on the location. It is the chain's first rollback of prices in its 11-year history and comes four days after McDonalds price cuts.

Locally, Roy Rogers will reduce the cost of its quarter-pound burgers by 4 cents in the District and 6 cents in the suburbs and Baltimore.

The company also said it will begin using 11 percent plumper birds in its fried chicken, without an increase in price. Prices of roast beef sandwiches and other products will remain the same.

This past weekend McDonalds cut the price of its hamburger and cheeseburgers by a nickel, its first cutback in 23 years. The chain had raised its prices last March boosting a 40-cent hamburger to 43 cents and a 45-cent cheeseburger to 48 cents.

The cut announced on Sunday would reduce hamburgers to 38 cents and cheeseburgers to 43 cents. Big Macs and Quarter Pounder sandwiches will not be reduced.

Despite Roy Rogers and McDonalds fighting it out, the expected Burger War of 1979 has apparently fizzled. The top burger chains said yesterday in interviews that they will not cut prices -- basically, they said, because the leader is not cutting prices on burgers of sizes comparable to theirs.

"I don't know what all the fanfare is about," said Burger King senior vice president Daryl Christensen. "We have a larger patty and a bun with sesame seeds on it," for around the same price as McDonalds regular hamburger, about 45 cents for a hamburger and 55 cents with cheese.

Burger King, ranked third in fast food revenues, does not plan to cut the price of its burgers, Christensen said.Wendy's and Hardees numbers four and five, aren't cutting prices either, spokesmen said.

"There's a question of motivation here," said Ginos vice president Rufus Schriber. "Why did they lower the prices just on the small burger? There's a question of long term benefit to consumers."

McDonalds spokesman Doug Timberlake said yesterday that their price cut was not to fuel a burger war. "Our price reduction was based on the beneficial commodity situation . . . we were able to pass along a savings to the customer. There was no consideration of what effect it would have on the competition or what the competition would do."

Why was just the small hamburger singled out for a price cut? "That's our basic product," Timberlake said. "We couldn't make it across the board."

What sparked the price decrease was the leveling off of beef prices and other commodities, Timberlake said.

An Agriculture spokesman confirmed that beef prices at the farm value have been declining by about 10 percent from April to last month. Increased cattle supplies is the reason, the spokesman said, and the decline is expected to last for several more months. In addition, demand may lessen with a recession thus reducing prices further, the spokesman said.

Although McDonalds' Timberlake said that prices of other commodities have also declined, giving the chain further reason to reduce prices, an Agriculture spokesman said that is not so. Since March the farm value measure used by the government has declined 5 percent from March to most recent figures for June, but wheat and dairy prices have been moving upward with the rate of inflation.

The agriculture spokesman also said that beef and other prices have declined significantly in the past, for instance during the last six months of 1975 and part of 1976, without a cut in prices at hamburger stands.