A proof of a conjecture of Erdős, Faudree, Rousseau and Schelp on subgraphs of minimum degree k

Abstract

Erdős, Faudree, Rousseau and Schelp observed the following fact for every fixed integer k≥2: Every graph on n≥k−1 vertices with at least (k−1)(n−k+2)+(k−22) edges contains a subgraph with minimum degree at least k. However, there are examples in which the whole graph is the only such subgraph. Erdős et al. conjectured that having just one more edge implies the existence of a subgraph on at most (1−εk)n vertices with minimum degree at least k, where εk>0 depends only on k. We prove this conjecture, using and extending ideas of Mousset, Noever and Škorić.