Vacancies are up 4.6 per cent on 2012, and have grown by 18.6 per cent in the public sector, despite deep cuts in spending planned for the next three years.
The Graduate Market in 2013 also finds that a quarter of the 100 organisations surveyed require a minimum Ucas tariff from applicants, usually based on their A-level grades.
Three quarters would not hire a graduate with less than an upper-second-class degree.
Despite an increase in vacancies, the median starting salary is £29,000, unchanged for the fourth year in a row.
Previous editions of the survey found that the number of openings plummeted by 17.8 per cent at the beginning of the recession in 2009, recovered in 2010 and 2011, and then dipped very slightly in 2012.
The rise in vacancies this year is unexpected, as earlier this year employers predicted they would increase by just 2.7 per cent compared to 2012.
On average there were 46 applications for each vacancy, with competition highest at consumer goods manufacturers, media organisations, banking and finance employers, retailers, and oil and energy companies. It was lowest in accountancy and professional services firms.
The best paid sectors were investment banking, oil and energy, law and consulting, all paying a median average starting salary of at least £30,000.
The public sector was the least well remunerated, paying graduates an average of £22,000 when they start.