Voters in this energy boom town, the nation's fifth largest city, will end a bitter mayoral runoff campaign Tuesday by choosing an arch-conservative former district attorney or a moderately conservative homebuilder and former city councilman.

The candidates, former DIstrict Attorney Frank Biscoe and former Councilman jim McConn, on the surface are in remarkable agreement in their approaches to the major issues, among them transportation, crime and the police department. However, the campaign has become a mud-slinging battle of personalities, with Briscoe attacking McConn as an inept businessman who will burt Houston's high municipal bond rating and McConn and others picturing Briscoe as divisive, a racist and a rumor monger.

Briscoe, 51, led a field of a dozen candidates Nov. 3 with 32 per cent of a sparse vote. A cousin of Texas Gov. Dolph Briscow, he is making his second race for mayor and has not held office since 1966, when he resigned as district attorney to make an unsuccessful bid for Congress.

Before the mayoral runoff, almost his entire campaign was restricted to television commercials and billboards, and he made few campaign appearances. Since then, he has appeared in public more frequently and refers repeatedly to McConn as "my bankrupt opponent - an allusion to the home-builders' acknowledged business debt of $412,000. Briscoe repeatedly suggests that "if McConn runs the city like he runs his business" Houston is in for financial ruin. Briscoe also recently called a press conference at which he introduced and cross-examined one of McConn's creditors.

McConn, 49, received 22 per cent of the vote Nov. 8 but subsequently picked up the endorsement of the fourth place contender, former television announcer and three-time mayoral candidate Dick Gottleib, who won about 19 per cent of the vote. The third place candidate, political novice Nobel Ginther, received 20 per cent of the vote, much of it from blacks and liberals. He has endorsed no one.

McConn said his debt was acquired during 1974 when the housing market collapsed and says he did not choose bankruptcy but rather is trying to pay off his creditors.

He has let others make the most serious charges against Briscoe. For example, the Black Organization for Leadership and Development, one of two major black groups endorsing McConn in the turnoff, says Briscoe was a racist who persecuted blacks when he was district attorney - a charge Briscoe denies.

Perhaps more serious was lame-duck mayor Fred Hofheinz's charge last week that Briscoe had planted rumors, which were investigated by a grand jury, that the 39-year-old mayor had been arrested during a drug raid, engaging in a homosexual act.

The extraordinarily widespread rumors, which were circulated last spring before Hofheinz announced he would not seek a third term, eventually generated sympathy for Hofheinz. The grand jury later reported that it had found no evidence of any criminal act.

Briscoe denied that he had started the rumor and contended that McConn "orchestrated" the mayor's attack against him. McConn denied that but he had earlier called a press conference to replay television newsfilm from last summer showing Briscoe responding testily and angrily stalking away when asked whether he was involved in spreading the rumors. The film has helped McConn portray Briscoe as a hot-headed, secretive man who would operate a "closed" administration.

Briscoe strategists have done their best to moderate his strong law-and-order image, much in evidence in his last mayoral campaign. One of Briscoe's few direct appeals to his traditional right-wing constituency was his appearance Saturday as a speaker before 11,000 opponents of the National Women's Conference here.

Voters also will fill the second highest city job, city comptroller from a race that pits a feminist Democrat accountant against a conservative Republican. Kathryn J. Whitmire, 31, won nearly 31 per cent of the general election vote to 21 per cent received by opponent Steve Jones.