Chacko sent his resignation to party president Sonia Gandhi and vice president Rahul Gandhi.
Maken said he will not take up any post in the party for a year and will perform his duty as an ordinary worker, while Chacko called for introspection to ascertain the reasons for its defeat in Delhi and redefine its future strategy.
"As President of Delhi Congress I take moral responsibility and offer to resign from my post. I have decided to resign from my post," Maken said.
Chacko on his part when asked told PTI, "Failure is failure and we must take responsibility".
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"Congress has to introspect. There are many reasons for the failure of the party and they should redefine their strategy for the future and that can be done by a new team and that is why we sent in our resignations," he said.
Maken claimed there has been a rise in the vote percentage of the Congress in Delhi.
His remarks came after former Delhi Chief Minister Sheila Dikshit today said that Maken failed to reach out to the people and even blamed his "lack of involvement" of the local leadership.
At the AICC briefing, party spokesperson Ajoy Kumar complimented Maken for his efforts at improving the party's vote share from nine to 21 per cent in the civic polls.
"Delhi Congress in-charge P C Chacko and state chief Ajay Maken are good, mature leaders and have worked very hard. Our vote share in Delhi has improved from 9 per cent in 2014 to 21 per cent today," he said defending Maken.
Kumar said "Maken and his team in Delhi Congress needs to be credited for improving the party's vote share from 9 per cent in Vidhan Sabha polls to 21 per cent now".
Congratulating the BJP, Kumar hoped it will give a new MCD which is free from corruption and is development- oriented.
He also said there was need for introspection within the Congress party.
"We will introspect on the reasons for the defeat. However, we have traversed a successful journey from 2015 assembly polls," he said.
"Congrats to BJP for winning MCD. Thank Congress workers for increase in vote share from 9.6 pc to 20 pc. Will work hard for a new narrative.
"Hope BJP will now work for a clean, disease-free, corruption-free, development-driven Delhi, unlike in past 10 years of its MCD rule," he said in a series of tweets.
Maken was appointed as Delhi Congress chief following the resignation of Arvinder Singh Lovely, in the backdrop of the party's rout in 2015 Assembly elections in which it drew a blank.
He served on the post for over two years and so did Chacko as in-charge of Delhi affairs.
In 2001, he became the Transport, Power and Tourism Minister in the Dikshit government and is credited with implementing the CNG-based public transport system and privatisation of power distribution in the national capital
At the age of 39 years, he became the Speaker of Delhi Assembly in 2004, earning the distinction of being the youngest person serving on the post in the country.
Steamrolling all opposition, the BJP today swept the MCD polls, trouncing the Aam Aadmi Party that rules Delhi, and decimating the Congress, in an election which was billed as a referendum on the Arvind Kejriwal government.
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