Long Cycles In Graphs Research Articles

Long cycles in graphs with no subgraphs of minimal degree 3

If a graph G has n vertices and 2 n−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 Erdős, 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).