Long Cycles in Graphs With no Subgraphs of Minimal Degree 3
If a graph G has n vertices and 2n – 1 edges, it must contain some proper subgraph of minimal degree 3. If G has one edge fewer and contains no such subgraph, then, as proved by Erdõs, Faudree, Gyárfás and Schelp, it contains a cycle of length at least [log n]. Our aim in this note is to prove an essentially best possible result, namely that such a graph must contain a cycle of length at least 4 log n +O(log log n). © 1989, Elsevier Inc. All rights reserved.
Annals of Discrete Mathematics
Bollobás, B., & Brightwell, G. (1989). Long Cycles in Graphs With no Subgraphs of Minimal Degree 3. Annals of Discrete Mathematics, 43 (C), 47-53. https://doi.org/10.1016/S0167-5060(08)70565-X