Bernicia
The first reliable report of Máel Coluim's reign is of an invasion of Bernicia in 1006, perhaps the customary crech ríg (literally royal prey, a raid by a new king made to demonstrate prowess in war), which involved a siege of Durham. This appears to have resulted in a heavy defeat by the Northumbrians, led by Uhtred of Bamburgh, later Earl of Bernicia, which is reported by the Annals of Ulster.
A second war in Bernicia, probably in 1018, was more successful. The Battle of Carham, by the River Tweed, was a victory for the Scots led by Máel Coluim and the men of Strathclyde led by their king, Owen the Bald. By this time Earl Uchtred may have been dead, and Eiríkr Hákonarson was appointed Earl of Northumbria by his brother-in-law Cnut the Great, although his authority seems to have been limited to the south, the former kingdom of Deira, and he took no action against the Scots so far as is known. The work De obsessione Dunelmi (The siege of Durham, associated with Symeon of Durham) claims that Uchtred's brother Eadwulf Cudel surrendered Lothian to Máel Coluim, presumably in the aftermath of the defeat at Carham. This is likely to have been the lands between Dunbar and the Tweed as other parts of Lothian had been under Scots control before this time. It has been suggested that Cnut received tribute from the Scots for Lothian, but as he had likely received none from the Bernician Earls this is not very probable.
Read more about this topic: Malcolm II Of Scotland