Cycles of consecutive lengths in $3$-connected graphs

Abstract

Recently Lin, Wang and Zhou have proved that every $3$-connected nonbipartite graph of minimum degree at least $k$ with $k\ge 6$ and order at least $k+2$ contains $k$ cycles of consecutive lengths. They also conjecture that this result is true for $k=4,5.$ We prove this conjecture. Our proofs use many ideas of Gao, Huo, Liu and Ma.

17 pages