Circumferences in 1-tough graphs Original Research Article

Bauer, Morgana, Schmeichel and Veldman have conjectured that the circumference c(G) of any 1-tough graph G of order n ⩾ 3 with minimum degree δ ⩾ n/3 is at least min{n,(3n + 1)/4 + δ/2} ⩾ (11n + 3)/12. They proved that under these conditions, c(G) ⩾ min{n, n/2 + δ} ⩾ 5n/6. Then Bauer, Schmeichel and Veldman improved this result by getting c(G) ⩾ min{n, n/2 + δ + 1} ⩾ 5n/6 + 1. We show in this paper that c(G) ⩾ min{n, (2n + 1 + 2δ)/3, (3n + 2δ - 2)/4} ⩾ min{(8n + 3)/9, (11n - 6)/12}.