Andrena Douglas, 53, was found dead in her house in Crusader Crescent, Stewarton, when officers went to the property at around 6.20am on Thursday.
They went there to tell Ms Douglas that her partner had been found seriously injured on the same morning after a fire at nearby Peacockbank Farm.
Police Scotland had been investigating whether the two events were connected and confirmed on Sunday that they now believe they are linked.
The force revealed on Saturday that it is treating Ms Douglas’s death as murder following the results of a post-mortem examination carried out the previous day.
A force spokeswoman said: “Police are treating her death as murder and inquiries are continuing.”
Officers have also released a photograph of Ms Douglas.
Detective Inspector Martin Fergus, of the major investigation team, previously said he wants to establish what happened between 8pm on Wednesday June 8 - when Ms Douglas was last seen - and the time her body was found the following day.
Urging anyone with information to come forward, he added: “To this end, I would appeal to anyone who was in the area, who may have seen any suspicious activity - whether that be a vehicle or someone on foot in the area - to contact police.”
He added: “How and why the fire started has still to be established and a joint investigation between police and Scottish Fire and Rescue is under way.”
Anyone with information about the murder or the fire is asked to contact the incident room in Kilmarnock on a new incident room number, 01563 505 172, or via 101.
Crimestoppers can be reached on 0800 555 111 where details can be given in confidence.