Graphs without proper subgraphs of minimum degree 3 and short cycles

Abstract

We study graphs on n vertices which have 2nź2 edges and no proper induced subgraphs of minimum degree 3. Erdźs, Faudree, Gyarfas, and Schelp conjectured that such graphs always have cycles of lengths 3,4,5,...,C(n) for some function C(n) tending to in finity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with n vertices and 2nź2 edges, containing no proper subgraph of minimum degree 3.