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Root went on to smash a masterly 142 not out, his 28th Test century, forging a mammoth 269-run partnership for the unbroken fourth wicket with Bairstow, who remained unbeaten on 114.
Bairstow said the verbal exchange was part and parcel of the game.
India finished day three at 125/3 after the hosts were bowled out for 284 earlier on the day.
Bairstow's gutsy 103 not out helped his side to 258/7 in reply to Australia's 416/8 declared, a position that looked highly unlikely when England limped battered and bruised to lunch on 36/4.
Aiden Markram has been named Malan's replacement in the Punjab Kings squad.
Rahul wasn't happy with the decision, pointing towards the back pad and shaking his head while walking back to the pavilion.
India were 43 for no loss in their second innings at stumps, still 56 behind.
"My phone is on and if he does want to give me a call or a message, he should feel free.," said the England ODI opener.
Mumbai will take heart from the fact that they defeated defending champions Chennai Super Kings in the last match.
Onmanorama picks five combinations which could make a huge difference to the campaigns of their respective sides.
Results 11-20 of 26
