The petition, filed by DMK working president M K Stalin, also wanted the floor test to be held under the supervision of an independent observer appointed by the high court.
It also sought a direction to the chief minister, refraining him from taking any major policy decision, pending the disposal of the petition.
The petition stated that 19 MLAs, loyal to AIADMK deputy general secretary TTV Dhinakaran, who is sidelined within his own party, had written individual letters to Governor Ch Vidyasagar Rao on August 22, informing him of their withdrawal of support to chief minister Palaniswami.
Thus, it was the duty of the governor to ask the government to prove its strength in the House, it said.
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"Exercising such a power is not a privilege but a matter of performance of official duty. By not calling for a trust vote, the governor has clearly failed in performing his constitutional duty," Stalin alleged in the petition.
He further charged that the governor, "by abdicating his constitutional duties", was "clearly working in tandem with the political party in power and deliberately delaying ordering the trust vote".
He added that the governor was required to perform his constitutional duties in a fair and objective manner, in accordance with the provisions of the Constitution and law.
Stalin had, on Sunday, called on the governor and urged him to direct the Palaniswami government to prove its majority in the Assembly within a week, in the wake of the revolt by the 19 AIADMK MLAs against the chief minister last month.
He had said that if the governor did not act on the DMK's plea within a week, the party would take legal recourse and also approach the "people's court".