Police said that Christopher Nicol, who was just 27, was stabbed to death in a 'brutal attack' at his home in Maple Road in Greenock last Thursday.
Officers believe that the motive for the murder was money and that Mr Nicol was targeted as the assailant believed that he had a large sum of money in his family home, which police have confirmed was not the case.
Neighbours have claimed that the killer apologised to Mr Nicol's girlfriend following the attack.
One told the Scottish Sun: "His girlfriend said the man said ‘give us the money’ and she just said ‘we didn’t have any’.
"She managed to grab the scarf off his face. She was shouting ‘can you please just go?’
“After the commotion she said he ran out the door and he said ‘sorry’ before locking the door shut.”
Mr Nicol's partner and neighbour tried to resuscitate him after the knife-man's attack, which took place as he was helping his children with a colouring book, police said.
DCI Martin Fergus said: "This was a young man who kept himself to himself and was a doting father to his children.
"At this time we believe that the motive for this was robbery and that Christopher was targeted specifically because his attacker thought there was a large sum of money in the house, which was not the case.
"He had been saving money to take his children to Florida. We believe the suspect knew Christopher and intended to rob him.
"For whatever the reason, to carry out such a brutal attack in front of such young children is sickening.
"It shows an absolute disregard for their safety or suffering. This callous killer must be caught."
The suspect is described as white, aged between 20 and 30, around 5ft 9in, of slim build, with an unkempt, reddish, brown beard and moustache.
He spoke with a local accent and had bad teeth with some visibly missing, and was wearing a black beanie hat with a logo, possibly Timberland, black top and black jeans or trousers or jogging bottoms.
Anyone with information is asked contact detectives from the Major Investigation Team (West) via 101 quoting incident number 3888 of September 26 2019 or contact Crimestoppers on 0800 555 111.