The two have been making such a demand ever since 18 disqualified AIADMK MLAs in September last year met then Governor C Vidyasagar Rao and told him that they had lost confidence in Chief Minister Palaniswami.
They had met Mr Rao a day after Mr Palaniswami merged the faction led by him with that of then rebel leader and now Deputy Chief Minister O Panneerselvam.
Mr Purohit who took over as the Governor in November last year, gave his maiden address to the state Assembly today, which was boycotted by the DMK-led opposition.
Mr Dhinakaran, however, sat in the House through the day.
DMK Working President MK Stalin, who is also the Leader of the Opposition, and Mr Dhinakaran, an independent MLA who called himself part of the opposition, reiterated their claim that Mr Palaniswami lacked the numbers to head the government.
They were apparently referring to the revolt of 18 MLAs who owed allegiance to Mr Dhinakaran and were disqualified by Speaker P Dhanapal last year.
Mr Stalin, who led an opposition-boycott of Mr Purohit's address, claimed it was "clear" that the government had the support of only 111 MLAs in a 235-strong House.
"Therefore, as there is a situation of a minority government (being in power) the Governor should have asked it to prove its majority. Rather, reading out an address prepared by the state government goes against democracy," Mr Stalin told reporters while explaining the reason for the boycott.
He also expressed his party's displeasure over the governor visiting various districts and holding meetings with state government officials, saying it was "against the Constitution."
Asked if the DMK would move a no-confidence motion against the government, Mr Stalin said he had already made it clear that he would do so if an opportunity arose. Right now, there is a court case in this, he said.
To a question on the possibility of working with Mr Dhinakaran, Mr Stalin said there was "no such necessity" for the DMK.
Mr Dhinakaran, who has given a jolt to the ruling AIADMK by defeating its candidate E Madhusudhanan in the December 21 RK Nagar by-poll, claimed that the government had only 111 MLAs on its side and he wants the governor to direct a floor test.
"As far as I know, there are only 111 (AIADMK) members supporting Palaniswami. For the governor, who is in a position to see if the chief minister has the majority, it is wrong on his part to have given the address as if endorsing this government," he told reporters.
Mr Dhinakaran claimed that the chief minister himself was "not hopeful" of the government fulfilling its term and said the ruling benches "concealed their own fears" by thumping desks in the House today.
He was apparently referring to cheering by AIADMK members whenever Mr Purohit mentioned Mr Palaniswami during his address.
The state government was "afraid" of the centre, Mr Dhinakaran claimed and described the governor's address as an "eyewash" and a "ritual."
He said subjects such as constituting Cauvery Management Board, safety issues pertaining to the Kudankulam Nuclear Power Plant in Tirunelveli district, post-Ockhi problems of fishermen and on the ongoing strike by a section of transport workers found no mention in the address.
Further, the governor's address did not mention the methane extraction project in the Cauvery delta districts, Mr Dhinakaran said, adding the initiative could harm the farmers.
Asked if he would support a no-confidence motion if brought by the opposition, mainly the DMK, against the government, he said that he had already announced extending support to such a move.
"There is nothing wrong in joining hands with the opposition to do good things," Mr Dhinakaran said.
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