For years, Richard Arrington has been viewed as the man most likely to become the first black mayor or Birmingham, and today the city's voters will decide whether he will.

But Arrington, a city councilman for eight years, is running for mayor far sooner than he thought, and the result is a campaign so close that while his opponent has a slim lead in the polls, the nonpartisan election is a tossup.

Paradoxically, the events that put Arrington 44, into the mayor's contest this year have also shaken a liberal white-black coalition that has been determining politics in recent years in this racially oriented, declining industrial city.

Public opinion over Arrington and his white opponent, Frank Parsons, is divided along almost solid racial lines, and, because whites make up more than half of the registered voters, Arrington needs the white support from the coalition to win today's runoff election.

Arrington, who received 44 percent of the vote in a crowded Oct. 9 election, has portrayed himself as the experienced candidate with the qualifications to be mayor. He has gained editorial endorsements from the city's two daily newspapers.

Parsons, who finished second with 17 percent, is a 38-year-old lawyer-businessman who worked on George Wallace's 1968 presidential campaign. lBefore white audiences he has referred to a "bloc vote" and has called law and order the city's No. 1 issue. He says the police department should be run by the police.

Those issues -- police conduct and law and order -- have been dominant in Birmingham since last June, when a white city policeman mistakenly shot an unarmed 20-year-old black woman to death.

Whites, according to a Birmingham Post-Herald survey, did not like a citizens review panel Mayor David Vann set up to look into the incident, balcks were unhappy that the policeman was not fired. And Vann, who was elected four years ago by the black and liberal white coalition largely put together by Arrington, finished a distant third on Oct. 9.

It was the shooting that prompted Arrington to split with his former ally, Vann, and run.

A poll taken by the Birmingham Post-Herald shows Parson with 44 percent and Arrington with 41 percent.