On a Conjecture of Woodall

Dirac proved in 1952 that every 2-connected graph of order n and minimum degree k admits a cycle of length at least {n,2k}. As a possible improvement, Woodall conjectured in 1975 that if a 2-connected graph of order n has at least n2+k vertices of degree at least k, then it has a cycle of length at least 2k. This conjecture was one of the 50 unsolved problems in Bondy and Murty (“Graph Theory with Applications,” Macmillan Press, New York, 1976). Häggkvist and Jackson showed in 1985 that this conjecture is true if n⩽3k−2. Häggkvist and Li proved that this result is true if the graph is 3-connected. In this paper, we essentially verify Woodallʹs conjecture.