It is also possible whoever was responsible fled to Northern Ireland.
Gardaí confirmed they are liaising with the PSNI.
There is as yet no known motive for what appears to be a random attack on the deceased who was married and had a number of children.
Mr Mulligan’s body was found 200 metres from his taxi on a small cross border road at Carnbeg, near Kilcurry, about 2.5km west of Dundalk.
The alarm was raised when three women, who were driving home in their own car, came upon the scene just after 3am.
Yesterday, Garda Superintendent Gerry Curley, Dundalk confirmed that Mr Mulligan was pronounced dead at the scene.
It is understood he looked as if he had suffered a number of stab wounds and that he may have tried to escape before collapsing onto the road.
It is not known if he was flagged down or if he had a passenger or passengers at the time.
Gardaí are also working to establish how the person, or persons, responsible left the scene afterwards.
A postmortem was carried out last night by deputy State Pathologist Dr Michael Curtis at Our Lady of Lourdes Hospital in Drogheda. He also visited the location where his body was found.
Superintendent Curley said: “I am sure there are people who know something about this incident.” He appealed for them to come forward.
The gardaí also want to hear from anyone who was a passenger with Mr Mulligan in his taxi on Sunday night. It was a beige/champagne colour Skoda Octavia registration number 07 LH 5599.
Yesterday, gardaí began to search the road and the land either side of it, including an adjacent large field.
Superintendent Curley said gardaí have established an incident room and both a senior investigation officer and a family liaison officer have been appointed.
House to house inquires are under way and gardaí are also examining CCTV from across Dundalk town to try and establish the movements of Mr Mulligans taxi during the night.
They have a number of sightings and these are being looked into.
He was an owner-driver and as such was self-employed and was not believed to have worked shifts for any particularly taxi company.
Superintendent Curley said: “We can confirm Martin started work as a taxi man around 9.30pm on Sunday and we are appealing to anyone who used this services after 9.30pm, or anyone who saw him or his car during the course of the night and anyone who was in the Carnmore /Kilcurry area between 1am and 3.30am to contact us.”
Mr Padraig Mulligan from Sevens cab company in Dundalk said: “Martin was a hard-working family man. He was very honest and was only trying to make a living.
“We are devastated and express our sympathies to his family,” he said.
The chairperson of Dundalk municipal district, Cllr Maria Doyle (FG) said: “This is a heinous crime and I hope that whoever was responsible will be brought to just as soon as possible. I would also urge anyone with any information to contact the gardaí.”