#### Title

Cycles and paths in graphs with large minimal degree

#### Abstract

Let G be a simple graph of order n and minimal degree > cn (0 < c < 1/2). We prove that (1) There exist n0 = n0(c) and k = k(c) such that if n > n0 and G contains a cycle Ct for some t > 2k, then G contains a cycle Ct-2s for some positive s < k; (2) Let G be 2-connected and non-bipartite. For each ε (0 < ε < 1), there exists n0 = n0(c, ε) such that if n ≥ n0 then G contains a cycle Ct for all t with [2/c] - 2 ≤ t ≤ 2(1 - ε)cn. This answers positively a question of Erdos, Faudree, Gyárfás and Schelp. © 2004 Wiley Periodicals, Inc.

#### Publication Title

Journal of Graph Theory

#### Recommended Citation

Nikiforov, V., & Schelp, R.
(2004). Cycles and paths in graphs with large minimal degree.* Journal of Graph Theory**, 47* (1), 39-52.
https://doi.org/10.1002/jgt.20015